claims depot shelf
The Riemann Hypothesis
Formal claims (Lean)
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The core claim is that the arithmetic spectrum of the Euler product, not the nontrivial zeros, carries the extensive thermodynamics of a black hole. The partition function $Z_{\mathrm{ar}}(\beta)=\prod_p(1-e^{-\beta\ln p})^{-1}=\zeta(\beta)$ describes free bosonic modes with energies $\ln p$; its linear entropy $S\simeq E/T_H$ with $T_H=1$ matches the linear-in-area entropy if the arithmetic cutof
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On the paper's own terms, the discovery is a fourth sum variant of the almost linear formula, built on the Gram sequence rather than on the zeros of $Z(t)=e^{i\vartheta(t)}\zeta(1/2+it)$. For each Gram point $t_n$ in the interval $(T,\varphi_1^{-1}(T))$, the signal $A_2(n)=[t_n,t_{n+1}]\times[0,Z^2(\tilde t(n))]$ has area $|A_2(n)|=Z^2(\tilde t(n))(t_{n+1}-t_n)$, where $Z^2(\tilde t(n))$ is the av
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The paper's central claim is a count rather than an existence: the number $a_{\omega,N}$ of even newforms of level $N$ whose $L$-function has at least one normalized zero in $(-\omega,\omega)$ obeys the lower bounds of Theorems 1.15 and 1.18. Theorem 1.15 says that if $\omega$ satisfies $-\bigl(\hat{\varphi}(0)+\tfrac12\int \hat{\varphi}(y)\,dy\bigr) < S(1,\sigma;\varphi_\omega)$, then along a seq
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The central discovery is that the Dyson class survives the nonlinear inverse-spectral map. For each unfolded spectrum $\{\varepsilon_n\}$, the dressing construction produces a deformation $f^{(s)}(x)$ of $H_0=-\frac{d^2}{dx^2}+\frac{x^2}{4}$; projecting onto the oscillator basis gives $F^{(s)}_{mn}=\langle m|f^{(s)}|n\rangle$, and the shell weights $W^{(s)}_d$ collect the matrix-element weight at
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The discovery, on the paper's own terms, is that two lower bounds for power sums can serve as the sole engine for zero-free regions: the first power-sum inequality detects the presence of an exceptional zero through the logarithmic derivative of a shifted Rankin–Selberg product, while a real-part power-sum bound, in combination with a zero-repulsion lemma, controls how close any other zero can com
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The discovery is the complete algebraic and dynamical classification of the contour Hankel matrices $H_m(\Gamma;\phi_\Gamma)$, whose entries are the contour moments $\mu_k=(2\pi i)^{-1}\int_\Gamma \phi_\Gamma(t)^k (\Xi'/\Xi)(t)\,dt$. By the residue representation, these moments are power sums over the coordinate images of the enclosed zeros, so $H_m$ factorizes as a Vandermonde–weight–Vandermonde
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The central discovery is that the weak interlacing relation between adjacent quotient indices is governed by a structural theorem for Jacobi spectral multipliers. For $0<\nu<2$ and $n\ge1$, define $h_{n,\nu}=\min\{1,\sqrt{1/4+e_{n,\nu}}\}$ with $e_{n,\nu}=n(n+1)(2-\nu)/(2n+\nu)$. If $F$ is a real even polynomial or entire function of order at most one with $Z(F)\subseteq S_{h_{n,\nu}}$, then every
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The central discovery is that the weak interlacing relation between adjacent quotient indices is governed by a structural theorem for Jacobi spectral multipliers. For $0<\nu<2$ and $n\ge1$, define $h_{n,\nu}=\min\{1,\sqrt{1/4+e_{n,\nu}}\}$ with $e_{n,\nu}=n(n+1)(2-\nu)/(2n+\nu)$. If $F$ is a real even polynomial or entire function of order at most one with $Z(F)\subseteq S_{h_{n,\nu}}$, then every
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The central claim is Theorem 1.1: for an absolute constant K > 0, the condition $n^{3}$ $log^{2}$(n+2) ≥ K $d^{5}$ forces all d zeros of $J^{{d,n}}$ to be real, distinct, and negative. The proof constructs a comparison polynomial with explicit simple positive roots, matches the first five normalized coefficients exactly, and then shows that the remaining coefficient multiplier is a small holomorph
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The paper's central claim is Theorem 1: assume RH and that there is an exceptional sequence of Siegel zeros $(\beta_j,\chi_j,q_j,E_j)$ with $q_j\to\infty$ and $E=\inf_j E_j$ sufficiently large; then $\mu=\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/(2\pi)<0.4733$. The proof chooses the resonator $R(t)=\sum_{n\le L}\chi(n)G(n)n^{-1/2-it}$, where $\chi$ is the exceptional quadratic char
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The central discovery is that the one-level density of the family $\{L(s,E_d)\}$ for $d$ odd fourth-power-free obeys, on average, the formula $D_{\mathcal F^*}(\phi,w,D)=\hat\phi(0)+\frac12\int_{\mathbb R}\hat\phi(u)\,du+O(1/\log D)$, where the Fourier support of $\phi$ is contained in $(-3/5,3/5)$ under GRH and in $(-1,1)$ under a quartic Patterson conjecture. This matches the expected Katz–Sarna
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Under RH for $\zeta(s)$ and $\beta(s)=L(s,\chi_{-4})$, the paper proves that for fixed $k\ge 2$ the signed count $D_k(x)=\sum_{n\le x,\Omega(n)=k}\chi(P^+(n))$ satisfies $$D_k(x)=-\frac{$x^{{1-1/(2k)}}$}{(\log x)^k}\left\{B_k+2\Re\sum_{\gamma>0}m_\rho C_k(\rho)$x^{{i\gamma/k}}$\right\}+O_k\left(\frac{$x^{{1-1/(2k)}}$}{(\log x)^{k+1}}\right),$$ where the sum over the zeros $\rho=\frac12+i\gamma$ of
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The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ fo
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The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n f
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Under the Deep Riemann Hypothesis, for any fixed spectral scale $T$ the difference between the mollified prime-power sums for $1$ and $a$ (mod $N$), normalized by $\log x / \sqrt{x}$, equals $C_N \log L(1, \chi_{1,a}) + O((\log x)/\sqrt{x})$. The leading growth and the principal-character noise cancel identically because the virtual character $\chi_{1,a}$ has coefficient $1 - \chi_0(a) = 0$, leavi
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The paper establishes two new results. First, Proposition 7.5: for R = X^ϑ with 0 < ϑ < 4/9, the smoothed Goldbach count r_φ(N) satisfies Σ_{N≤X} |r_φ(N) − N S(N) − M(N;R) − Z(N;R)|² ≪ (X^{3−ϑ} + X^{13/5})(log X)^5, where M and Z collect, explicitly, the contribution of every zero of every Dirichlet L-function of conductor at most R. Second, Theorem 8.2: if for some A > 5/2 and δ ∈ (0,1) the spars
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The central object is the Dirichlet series F_ℓ(s) = ∑ φ_ℓ(n)n^{-s} = ζ(s−1)/ζ(ℓ(s−1)+1). Its nontrivial poles occur at s = 1+(ρ−1)/ℓ for each nontrivial zero ρ of ζ, so the location of zeta zeros is encoded in the poles of F_ℓ. The paper's key analytic step is to write Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + E_ℓ(x) and form the Mellin-type integral G(s)=∫_1^∞ E_ℓ(x)x^{-s−1}dx, which equals F_ℓ(s)/s − (1/(2ζ(ℓ+1))
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Theorem 1.1 states that, assuming GRH, for sufficiently large X, max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a resonator R_q = Σ_{m∈M} χ_{8q}(m), with M a set of N = X^{1/4-δ} square-free integers chosen to maximize the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). The core of the argument is the ratio S2/S1: S1 counts the diagonal terms m=n via a conditiona
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Equation (1.1) states that for distinct primitive characters χ0 and χ1 with moduli at most Q, every integer x ≥ 10, every sufficiently large T, and every window width Δ between T exp(−C√log T) and T/log T, the sum over zeros ρ = β + iγ of L(s, χ0) with T−Δ < γ ≤ T of x^ρ L(ρ, χ1) equals (χ1(x)/(2π)) Δ log T + O(Δ). This is the first short-window version of such moments, reaching windows shorter th
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At the top of the paper stands Theorem 1.1: assuming GRH and D^272 ≪ q^{11/16-1/2000}, for any four intervals U_1,...,U_4 the weighted proportion μ_F of primitive characters χ mod q for which log|L(1/2,χχ_j)|/sqrt(1/2 log log q) ∈ U_j for all j equals the four-dimensional Gaussian integral 1/(4π^2)∫_{U_1×...×U_4} e^{-Σ x_j^2/2} dx, up to O_ε((log log q)^{-1/2+ε}). In other words, the four normaliz
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The central discovery is Theorem 1.6: for any odd prime p > 3, the number f₂(p) of k in the half-interval 1 ≤ k < p/2 with {k²/p} > 1/2 is exactly (p−1)/4 when p ≡ 1 mod 4, (p−1)/4 − (3/2)h(−p) when p ≡ 3 mod 8, and (p−1)/4 − (1/2)h(−p) when p ≡ 7 mod 8, where h(−p) is the class number. Combining this with Dirichlet's class number formula and an unconditional lower bound for L(1, χ_p), the authors
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On the paper's own terms, the central discovery is a structural decomposition of a Hilbert–Pólya-type operator. The small-aperture spectrum is a universal Archimedean ladder: $\lambda_k(a)=\log(1/a)+\log(k-\frac12)+B_0+O(a)$ with $B_0=\log q-2\log 2$, where the constant emerges from cancellation rather than fitting, and the degree of the L-function appears as the logarithmic slope of the ladder. T
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The central claim is Theorem 1.1: for a smooth compactly supported test function Φ and any fixed scale y>0 and window exponent δ∈(3/4,1), the scaled average of quadratic Hecke characters over primes of the Gaussian integers converges as X→∞ to M_Φ(y,δ) = (1/4) Σ_{l primary} μ[i](l)/N(l²) Σ_{k∈O_K, k≠0} (−1)^{N(k)} Φ̃(N(k)√(1/(2y N(l²)))), where 'primary' means congruent to 1 modulo (1+i)^3 in Z[i]
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For 1<α_1<...<α_r, the vector (log ζ(s+i(log τ)^{α_1}),...,log ζ(s+i(log τ)^{α_r})) is jointly universal in the strip x_Φ(α_1)<σ<1, and under the Riemann hypothesis in the whole strip 1/2<σ<1. In the same range, the discrepancy between the empirical distribution of log ζ(σ+i(log t)^r) and the random Euler product limit is O(((r-1) log log T)^{-σ}). The paper also proves a unified discrepancy estim
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For 1<α_1<...<α_r, the vector (log ζ(s+i(log τ)^{α_1}),...,log ζ(s+i(log τ)^{α_r})) is jointly universal in the strip x_Φ(α_1)<σ<1, and under the Riemann hypothesis in the whole strip 1/2<σ<1. In the same range, the discrepancy between the empirical distribution of log ζ(σ+i(log t)^r) and the random Euler product limit is O(((r-1) log log T)^{-σ}). The paper also proves a unified discrepancy estim
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The central claim is that the BBM similarity structure fixes a metric topology that is fatal to the original Hilbert–Pólya mechanism, even though it supports many operator realizations. The positive form $\eta_0=\Delta^\dagger\Delta=2I-\hat S-\hat S^\dagger$ is not coercive: normalized Cauchy-kernel probes concentrated near $k=2\pi$ have metric length $\langle\phi_\varepsilon,\eta_0\phi_\varepsilo
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Lemma 1 is the load-bearing identity: for T→∞, Σ_{T<γ_n<¹T} ∫_{γ_n}^{γ_{n+1}} Z(t)² dt = (1−c)T + O(√T), where ¹T=φ₁^{-1}(T) is the first reverse iteration of Jacob's ladder and γ_n are the zeros of Z(t) on the critical line. The paper derives it by partitioning the integral over (T,¹T) into integrals over consecutive zero-to-zero intervals, using an elementary bound on the edge gaps. Substituting
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The central claim is that D_{r,k}>0 for every r≥2 and every k≥10^18 r^3. The author establishes this through a chain: a Cauchy–Schwarz plus Turán argument pins the curvature τ_k between 1/(2k) and 4/k; a certified saddle analysis of I(z)=∫u^{2z}Φ(u)du on relative disks |z−k|≤0.05k gives the zero-free factorization I(z)=e^{Ψ_z(u_s)}√(2π/(−Ψ''_z(u_s)))(1+ε(z)) with |ε|<0.018, and hence the uniform a
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On the paper's own terms, the central discovery is that the pair-correlation function F^+_{χ_□}(x,T), which sums x^{i(γ_1−γ_2)} over pairs of zeros of the quadratic character L-function weighted by W(γ_1−γ_2)=4/(4+(γ_1−γ_2)^2), satisfies F^+ ≪ T log(kx) for x ≤ T ≤ e^x (Theorem 1) and, under GRH, has the Montgomery-type asymptotic F^+ ∼ (T/2π) log x (Theorem 2). The paper then postulates (Hypothes
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The central discovery is an exact finite-scale decomposition of the remainder. For 0<u<=1/2, identity (9.36) writes E_rho3(e^{-u}) as log(1/u) A_rho3(u^{-1}) + C_rho3(u^{-1}) + U_rho3(u) + V_rho3(u), where A and C are finite sums over coprime pairs with n+m <= u^{-1} of the Möbius-weighted remainder times a density factor, U is the corresponding dilation-error sum, and V is the far tail. Corollary
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Under the assumption that every nontrivial zero of ζ is simple, the paper proves the exact identity Φ(e^{-x}) = Σ_ρ Γ(ρ)x^{-ρ}/ζ'(ρ) + πκ(x) + 2λ(x) + 4υ(x) − 2β(x)log(2πx) − 2. The distinguishing structural feature is that Γ(s)x^{-s}/ζ(s) has double poles at the negative even integers, where poles of Γ collide with trivial zeros of ζ; their residues contain the logarithmic term and the digamma an
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Under the assumption that every nontrivial zero of ζ is simple, the paper proves the exact identity Φ(e^{-x}) = Σ_ρ Γ(ρ)x^{-ρ}/ζ'(ρ) + πκ(x) + 2λ(x) + 4υ(x) − 2β(x)log(2πx) − 2. The distinguishing structural feature is that Γ(s)x^{-s}/ζ(s) has double poles at the negative even integers, where poles of Γ collide with trivial zeros of ζ; their residues contain the logarithmic term and the digamma an
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The exact maximal reflection-invariant zero region that forces the centered binomial sample of a balanced entire function of order at most one to lie on the unit circle is the hyperbolic region Omega_d. Once zeros lie in a strictly thinner strip, consecutive derivative samples have only simple unit-circle zeros that strictly cyclically interlace, provided a single non-vanishing finite-difference c
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The Hadamard–Weierstrass factorization of the entire function ξ already implies that every non-trivial zero of the Riemann zeta function lies on the critical line Re(s)=1/2.
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Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed simple critical-line zero, the full Gaussian–Perron prime-force defect near that zero equals the universal selected-zero profile −a Re(e^{−λ}/λ) plus an error that is O(1/log X) plus exponentially small, uniformly on compact sets of the logarithmic displacement λ away from zero.
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The central claim is Theorem 1.1: for a finite Galois extension K/Q, under GRH for ζ_K, and for any positive exponents a_j and shifts b_j with |b_j|≤T/2, the shifted moment ∫_0^T ∏_{j=1}^{2k} |ζ_K(1/2+i(t+b_j))|^{a_j} dt is bounded by T (log T)^{[K:Q](a_1^2+...+a_{2k}^2)/4} times a product over pairs of the correlation function g(|b_i-b_j|)^{[K:Q] a_i a_j/2}. Setting all a_j=1 and b_j=0 gives the
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Assuming the Riemann hypothesis, the two damped nets are moderate, uniformly L²-bounded, and associated with −χ_{(0,1)}; conversely, the mere existence of any moderate net of this damped-Báez–Duarte form that is uniformly L²-bounded and associated with −χ_{(0,1)} forces the Riemann hypothesis.
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Assuming the Riemann hypothesis, the two damped nets are moderate, uniformly L²-bounded, and associated with −χ_{(0,1)}; conversely, the mere existence of any moderate net of this damped-Báez–Duarte form that is uniformly L²-bounded and associated with −χ_{(0,1)} forces the Riemann hypothesis.
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The paper proves that under GRH the Linnik–Goldbach problem can be solved with K=6, meaning every sufficiently large even integer is p1+p2+2^a1+...+2^a6, improving the previous K=7. Unconditionally, it proves that the lower density of numbers representable as p+2^a is at least 0.12532, so at least a quarter of odd numbers are of this form. Both improvements follow from replacing the classical siev
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The universal late-time attractor of freely decaying incompressible turbulence is the planar Euler ensemble of rational star-polygon walks. Its continuum limit produces dimension-independent energy scaling functions whose Mellin amplitudes contain the Riemann-wall poles p = –8 + iρ_n generated by the non-trivial zeros of ζ(s). Assuming the Riemann Hypothesis and simplicity of those zeros, the pole
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The universal late-time attractor of freely decaying incompressible turbulence is the planar Euler ensemble of rational star-polygon walks. Its continuum limit produces dimension-independent energy scaling functions whose Mellin amplitudes contain the Riemann-wall poles p = –8 + iρ_n generated by the non-trivial zeros of ζ(s). Assuming the Riemann Hypothesis and simplicity of those zeros, the pole
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The universal late-time attractor of freely decaying incompressible turbulence is the planar Euler ensemble of rational star-polygon walks. Its continuum limit produces dimension-independent energy scaling functions whose Mellin amplitudes contain the Riemann-wall poles p = –8 + iρ_n generated by the non-trivial zeros of ζ(s). Assuming the Riemann Hypothesis and simplicity of those zeros, the pole
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Theorem 1.3: For fixed q ≥ 2 and k ≥ 2, under GRH for Dirichlet L-functions modulo q, the average G_{q,k}(N) satisfies G_{q,k}(N) = Σ_k(q)/φ(q) · G_{1,k}(N) + O(N^{1/k} log^2 N log q / φ(q)), where Σ_k(q) = Σ_{χ^k = χ_0} χ(−1). The main term is therefore a constant multiple of the unrestricted average G_{1,k}(N), and the constant is an explicit character sum that can vanish. When Σ_k(q) = 0, the a
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For every integer n greater than or equal to zero the regular coefficient Cn in the Laurent series of the secondary zeta function about s=1 is recovered by the limit formula Cn = lim (T→∞) (-1)^n {sum_{γ<T} log^n(γ)/γ - [1/(2π(n+1)(n+2))] log^{n+1}(T) log(T^{n+1}/(2π)^{n+2})}.
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Informational cardinality I(M)=(α(M),δ(M),ι(M)) distinguishes the essential fractal prime set P_ess from the generalized Cantor set C_{1/3}: both have continuum cardinality (α=1), yet dim_H(P_ess)=1/2>1/3=dim_H(C_{1/3}) and ι(P_ess)=-ζ(1/2) while ι(C_{1/3})=0, so I(P_ess)>I(C_{1/3}) under lexicographic order. The same framework pairs P_ess with a fractal zero set Z_F of equal dimension and conject
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The central claim is Theorem 1.1: under the generalized Riemann hypothesis and the negative-moment bound ∑_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (or the corresponding L-function version for odd k), the function y ↦ e^{-y/(2k)}∑_{n≤e^y} f(n) has a limiting distribution ν_k on R. Theorem 1.2 strengthens the earlier conjecture: except on a set of finite logarithmic measure, ∑_{n≤x} f(n) ≪_ε x^{1/(2k)} (l
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The central discovery is that the Fourier-Laplace transform along a vertical line, with the Gaussian kernel, yields a globally convergent, Gaussian-damped integral for the reciprocal Gamma function: $G(z)=\int_{-\infty}^{\infty} w^{2z} e^{w^2} \, dt = \pi/\Gamma(1/2-z) = \cos(\pi z)\Gamma(z+1/2)$, valid for all $z\in\mathbb{C}$. The scaling identity $G(z,\alpha)=\alpha^{-(z+1/2)}G(z)$ turns each D
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The paper establishes explicit two-sided bounds for Re ζ'/ζ(1+it) under the Riemann hypothesis, for all t ≥ e^18. As consequences it obtains: |ζ'/ζ(1+it)| ≤ 2 log log t + 0.0784 − γ + 9.0581 log log t/log t − 4.7/log t (so that in particular |ζ'/ζ(1+it)| ≤ 2 log log t for t ≥ 10^30); |ζ(1+it)| ≤ 2e^γ( log log t − log 2 + 1/2 + 0.2674/log log t − 2.676 log log t/log t ); and 1/|ζ(1+it)| ≤ (12e^γ/π^
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The central claim is that the full weight bound for all separated finite-type schemes over $\mathbb{F}_q$ follows once the middle cohomology of a singular affine hypersurface is controlled. The paper controls it by interpolating between the given singular hypersurface and a smooth one through a one-parameter family, transferring the known weight bound from the smooth fibers to the special fiber vi
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In the paper's own terms, the central discovery is the bound (1.2.1): under the Riemann Hypothesis, for every 0<δ<1 and x≥e^e, |E^AN_{σ1}(x)| ≪ x^{δ'} exp(log x / log log x) with δ'=max{1/2,δ}. Here E^AN_{σ1}(x) = 1/2∑_{n≥1}{x/n}² + x/2(log x+2γ−1) is the analytic part of the error term E_{σ1}(x)=∑_{n≤x}σ_1(n)−(π²/12)x², obtained through the Volterra integral equation decomposition. The proof deri
Stated claims
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The paper claims to have proven the Riemann Hypothesis: all nontrivial zeros of ζ(s) lie on the line Re(s)=1/2. The central result is stated as Theorem 1.1, and the proof is presented in Section 3 through Propositions 3.1 and 3.2.
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The paper establishes that the Riemann hypothesis is equivalent to the condition that for every real admissible pair of smooth functions (f,g) — supported respectively on [0,w(L)] and [L-w(L),L] with log(1/w(L))=o(L) — the normalized Weil cross term R(f,g) ≤ 1 for all sufficiently large L. If RH holds, Cauchy-Schwarz on the zero sequence gives the bound directly. If RH fails, a zero with real part β>1/2 forces the existence of pairs with R(f,g) > $e^{{ηL}}$ for any η < β-1/2, violating the bound. The proof combines a short-support positivity estimate (Lemma 1), a smooth partition procedure that transfers a bound on narrow-supported pairs to translates of a fixed function with a subexponential cost (Lemma 2), and a Laplace-transform pole argument showing that an exponential bound on the translated term would contradict the presence of an off-line zero (Lemma 3). The width condition log(1/w(L))=o(L) is essential: it ensures the partition cost remains subexponential, so the exponential violation from a false zero cannot be absorbed.
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At the center stands Theorem 1: every number field $K$ of odd degree $n \ge 3$ and absolute discriminant $D_K$ possesses an integral polynomial $f(x) = a_0x^n + \cdots + a_n \in \mathbb{Z}[x]$ of degree $n$ with a root $\alpha$ such that $K = \mathbb{Q}(\alpha)$ and $H(f) = \max |a_i| \ll_n D_K^{(n-1)/(2n-1)}$. This sharpens the square-root bound $O_n(D_K^{1/2})$ that the authors' earlier work established for all degrees, and it is a genuine improvement because the exponent $1/2$ is known to be optimal in even degrees, the obstruction coming from towers of quadratic extensions and a relative discriminant inequality. The mechanism is lattice duality: a short integral generator produces a monic norm polynomial, whereas short elements of the inverse different produce the non-monic polynomial $D_K \cdot N_{K/\mathbb{Q}}(\alpha x - \beta)$, and the inequality relating successive minima of a lattice and its trace dual guarantees that one of the two constructions always succeeds; the real-embedding translation saves one factor, and the absence of an index-two subfield is what guarantees a generating ratio exists. Under a zero-free half-plane for the Dedekind zeta function, a degree-one prime at a suitable scale supplies the substitution $f(x) \mapsto f(px+r)/p$, which divides the height by $p$ and preserves the field, yielding the exponent $\rho(n,m(K))$ for fields of every signature, and in degree three the matching lower bound shows the optimal exponent is exactly $1/3$ along an infinite family of pure cubic fields.
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The central claim is Theorem 1.3: modulo Hypothesis 1.2, each of unknot recognition, 3-sphere recognition, torus knot recognition, hyperbolic knot recognition, and hyperbolicity of cusped 3-manifolds lies in coNP. The proof has two parts. First, following Kuperberg, each decision problem reduces to asking for a nonabelian representation $\pi_1(Y) \to \mathrm{SL}(2,\mathbb{C})$, which by Koiran's work is equivalent to finding a root of a single irreducible polynomial $f \in \mathbb{Z}[x]$ modulo some prime $p$. Second, the size of $p$ is controlled by the effective Chebotarev density theorem: the cited prior work [FSW26] proved that Hypothesis 1.2 implies an error term of size $c \cdot \frac{\#C}{\#G} x^{1-\epsilon} \log(\Delta_L^{1/\epsilon} x^{[L:\mathbb{Q}]})$ in counting primes of norm at most $x$. Because the discriminant enters only logarithmically and the zero-free region is uniform over all number fields with one fixed $\epsilon$, the authors read off that $x$ can be chosen polynomial in $d$ and $\log\|f\|$, proving Theorem 2.1. Theorem 2.1 then plugs into Kuperberg's framework verbatim, yielding Theorem 1.3 and the corollaries on nonabelian homomorphisms to $\mathrm{SL}(2,\mathbb{F}_p)$ and regular nonabelian covers of degree at most $e^{\mathrm{poly}(n)}$.
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The paper establishes that Λ < 156821/1000000 = 0.156821, and does so by excluding the possibility of a multiple real zero at any point in the time band [0.156821, 0.2] and spatial region |x| ≥ 6000000185827 ∪ [X,W] ∪ [W,∞) (with explicit constants). The argument separates three regions: a stationary separator at x = X, a compact interior where all zeros are real and simple by propagation, a finite exterior where five Gram sources guarantee a stable polynomial modulus > 1/790, and an infinite exterior where a differential inequality gives u² + u_x² > 3/10000. A critical-time compactness lemma shows that if Λ lies in the band, H_Λ would have a finite multiple real zero; the regional exclusions contradict that, forcing Λ below the band. The proof does not assume any new zero-free half-plane or unverified analytic hypotheses beyond the published Polymath approximations and Platt–Trudgian finite-RH result.
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For a specific one-dimensional diffusion with invariant law proportional to the kernel of Riemann's xi function, one can prove unconditionally that the variance under quadratic tilts obeys a strict chain 0 < χ_a(h) < m_a(h)/h < v_a, that the variance after an outward quench relaxes monotonically with an exponential envelope, and that all zero-field alternating cumulant signs are positive. The genuinely stronger structure—a representation of the logarithmic derivative as a Stieltjes transform of a positive measure and the consequent positive semidefiniteness of all Löwner matrices built from that function—holds unconditionally for a ≥ 0.05, but at a = 0 it is equivalent to the Riemann Hypothesis. The author treats the entire RH-conditional hierarchy as a mathematical equivalence rather than a claimed new reformulation.
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The central claim is a two-sided sandwich on a single threshold. On one side, Lemma 3.1 transfers Noor's disk-space convergence to the family $g_{k,\tau}(z)=k^{\tau-1}\sum_n z^n/n^\tau-\sum_n z^{nk}/n^\tau$, showing $\sum_k \mu(k)k^{-\tau}g_{k,\tau}\to -z$ in $H^2_0$ for each $\tau>1/2$, and a bounded operator $M_\tau:f\mapsto f(s-\tau)/s$ from the Hilbert space of Dirichlet series into $H^2(\mathbb{C}_{1/2+\tau})$ carries this to $S_n=\sum_{k=2}^n\mu(k)G_k\to E$ in $H^2(\mathbb{C}_\alpha)$ for every $\alpha>1$ (Theorem 3.2, Corollary 3.3). On the other side, because $G_k(\rho)=0$ at every zero of $\zeta$ and the space is a reproducing-kernel space, the evaluation functional at a zero $\rho$ gives $\|E-\sum c_kG_k\|_\alpha^2\ge |E(\rho)|^2/\|k_\rho\|_\alpha^2=2(\mathrm{Re}\,\rho-\alpha)/|\rho|^2$ for all coefficients, with a determinant form $v^*K^{-1}v$ for several zeros (Theorem 3.5). Hence the threshold $\alpha_c=\inf\{\alpha\ge1/2: d_n(\alpha)^2\to0\}$ satisfies $1/2\le\Theta\le\alpha_c\le1$, where $\Theta$ is the supremum of the real parts of the zeros, and $\alpha_c=1/2$ exactly under RH (Corollary 3.6). The numerical half rests on Proposition 4.1, which writes $s_n(\alpha)=\|S_n-E\|_\alpha^2$ as $\sum_{m\ge1}w_m(\alpha)R_n(m)^2$ with $w_m(\alpha)=(m^{-2\alpha}-(m+1)^{-2\alpha})/(2\alpha)$ and $R_n(m)=1+\sum_{k=2}^n\mu(k)(m\bmod k)/k$; computing this to $n=10^6$ shows $s_n(1/2)$ oscillating without decay, while for $\alpha>1/2$ the envelope falls like $n^{1-2\alpha}$, far faster than the conditional bound $n^{-(\alpha-1/2)/3}$ available under RH, and on the critical line the optimal error satisfies $d_n(1/2)^2\log n\approx0.046$ for $100\le n\le5120$, matching the constant $2+\gamma-\log4\pi$ and earlier computations.
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The load-bearing assertion is Theorem 3.1. Let $F(s) = \exp(\int_1^{\infty} x^{-s}\,dm(x))$ for a positive measure $dm$, continued analytically to the strip $\sigma \ge 1 - 1/\psi(|t|)$ except for a simple pole at $s = 1$, with the pole-side estimates $F(\sigma) \le A/(\sigma-1)$ and $-F'(\sigma)/F(\sigma) \le 1/(\sigma-1) + B$ for $1 < \sigma \le 2$, and with the uniform bound $|F(s)| \le e^{\varphi(t)}$ throughout the strip; here $\varphi, \psi$ are positive, non-decreasing, slowly varying, and $\psi(t) \ll e^{(\alpha+o(1))\varphi(t)}$. Then for every $c > 1/(2(1+\alpha)V)$ the function $F$ has no zeros in the region $\sigma \ge 1 - c/(\psi(t)\varphi(t))$, $t \ge t_0$, where $V$ is Landau's extremal constant from the theory of nonnegative cosine polynomials. The theorem works because the positivity of the measure, not a functional equation, drives the argument, so it applies exactly where the local method is needed: Beurling zeta functions, which possess no analytic continuation machinery beyond a strip. Applying it with $\psi(t) = 2/(1-\theta)$ and $\varphi(t) = (1/2+\varepsilon)\log t + O(1)$ to systems with $N(x) = Ax + O(x^{\theta})$ gives Corollary 3.3: no zeros for $\sigma > 1 - (1/(2V) - \varepsilon)(1-\theta)/\log|t|$, hence the prime number theorem with error $O(x\exp(-c'\sqrt{\log x}))$ for $c' < \sqrt{2(1-\theta)/V}$. The same hypotheses are then shown to control the local number of zeros with explicit constants, to extend, for almost any $b$, Littlewood's short-interval zero-counting under the Riemann hypothesis to settings without a functional equation, and to force any zero close to the 1-line to come with a cluster of nearby zeros; feeding the clustering information back into the main estimate shows the constant $1/(2W_\alpha)$ is not sharp.
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The central claim, Theorem 1.2, is that without assuming RH the densities $\Theta$ and $\Lambda$ are strictly log-concave and are P\'olya frequency functions of the explicit finite order $k_* = 1\,810\,540\,931\,075$: for every $k \le k_*$ and every increasing tuples $x_1 < \cdots < x_k$ and $y_1 < \cdots < y_k$, the determinant $\det[\Theta(x_i - y_j)]$ is nonnegative and is positive exactly when $x_i > y_i$ for all $i$, while $\det[\Lambda(x_i - y_j)]$ is always positive. Log-concavity, the order-two case, uses only zeros through height 100; the trillion-scale order uses the full published verification to height $3\,000\,175\,332\,800$ together with an analytic error bound that keeps the numerical part to scalar inequalities. The paper further proves that the reciprocal laws exist as positive self-decomposable probability laws with smooth densities, that $1/\xi(\tfrac12 + \|u\|)^\alpha$ is strictly positive definite on every $\mathbb{R}^d$, and a converse obstruction: if either density is PF$_k$, then every zeta zero up to the $(k+2)$-nd must lie on the critical line. Finally, if RH holds up to height $T > \tfrac12$, every generalized Laguerre inequality $L_n[\Xi](x) \ge 0$ holds on $|x| \le T - \tfrac12$, extending the first-order case to all orders; with the published verification this covers $|x| \le 3\cdot10^{12} - \tfrac12$.
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The central claim is Proposition 7.1: for $F = L(s,\Pi)$ assumed to factor as in (1.1) into inequivalent unitary cuspidal representations over $\mathbb{Q}$, and for a fixed unitary cuspidal probe $P = L(s,\pi)$, any nonzero, nonnegative, even Gevrey-order-two Fourier amplitude $\phi^*$ supported away from zero gives $$\frac{$R^{{\phi^*}}$_{FP,FP}(T) - $R^{{\phi^*}}$_{F,F}(T) - $R^{{\phi^*}}$_{P,P}(T)}{2J_{\phi^*}} = r_\Pi(\pi) + O($e^{{-a\sqrt{\log T}}$}),$$ where $r_\Pi(\pi)$ is the multiplicity of $\pi$ in the factorization and is zero when $\pi$ is absent. The identity behind the detector is exact at finite height — half the alternating sum of the three raw correlations equals the mixed correlation $R_{F,P}(T)$ — and the choice $\phi^*(0) = 0$ eliminates every smooth-density and finite-part term, leaving the overlap coefficient $r_{F,P}J_{\phi^*}$ plus a subexponential remainder produced by a local meromorphic argument near $s=1$. The paper further establishes a full expansion of raw mixed $n$-level correlations at every fixed level and inverse-logarithmic order, whose coefficients are archimedean terms, Laurent coefficients of mixed Rankin–Selberg series, and moments of connected prime-power cumulants; the first genuinely three-coordinate invariant $\Gamma_3$ enters at order $(\log T)^{-3}$ through relations such as $p^a p^b = p^{a+b}$. An explicit example with products of primitive Dirichlet characters modulo $13$ shows two $L$-functions with identical gamma factors and identical complete self-pair arithmetic series but different $\Gamma_3$ values, proving that level-three data carry arithmetic information invisible to every self-pair statistic.
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The central claim is Theorem 1.1: assuming GRH for Dirichlet $L$-functions and for finite-order Hecke $L$-functions over $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{-3})$, the smoothed first moment over admissible characters of exact order $j$ equals $C_j\hat{\Phi}(1)Q + O(Q^{\theta_j+\varepsilon})$ with $\theta_4 = 49/58$, $\theta_6 = 23/26$, and positive constants $C_4$, $C_6$ given explicitly in (3.46) and (4.32). The proof continues the double Dirichlet series $A_j(s,w)$ to a region whose boundary at $w=1/2$ is exactly $\mathrm{Re}\,s = \theta_j$, then shifts the Mellin contour across the single pole at $s=1$. The enabling step is a joint estimate for the two sums left after Möbius inversion removes rational prime divisors from the Gaussian or Eisenstein modulus: with $\mu(d)$ retained, the pair factors as a quotient $L(z,\chi)/L^{(2)}(t,\chi^{2})$ times a convergent correction product in the quartic case, and similarly with $\chi^{4}$ in the sextic case, so GRH bounds numerator and reciprocal simultaneously. Combining this with the twist-exponent estimate for the completed quartic Gauss-sum series (Proposition 2.1) and the sextic analogue, then interpolating by Phragmén–Lindelöf with continuation to the convex hull via Bochner's tube theorem, gives boundary lines $58\,\mathrm{Re}\,s + 40\,\mathrm{Re}\,w = 69$ and $26\,\mathrm{Re}\,s + 20\,\mathrm{Re}\,w = 33$. The main terms are the same as in the earlier work, so the improvement is confined to the error term.
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The claim, stated as Theorem 1.1, is that under GRH for Dirichlet L-functions and for Hecke L-functions over $K = \mathbb{Q}(\sqrt{-3})$, the smoothed first moment over primitive cubic characters satisfies $$\sum_{(q,3)=1}\;\sum_{\substack{\chi\bmod q\\ \$chi^{3}$=\chi_0}}^{*}\!\! L(\tfrac12,\chi)\,\Phi(q/Q) = C_3\hat\Phi(1)Q + C'_3\hat\Phi(\tfrac56)$Q^{{5/6}}$ + O_{\Phi,\varepsilon}($Q^{{23/28+\varepsilon}}$).$$ The leading constant $C_3$ is the one obtained by [5] at the central point; the new constant $C'_3$, displayed in (1.4), is a product of $(2\pi)^{5/3}$, $\zeta(3/2)$, $\zeta(5/6)$, a gamma factor divided by $\zeta_K(2)$, the ramified factor $1-3^{-5/3}$, and two absolutely convergent Euler products over primes $p\equiv 1$ and $p\equiv 2 \pmod 3$. Every factor is positive except $\zeta(5/6)<0$, so $C'_3<0$. The proof's content is analytic continuation: the double Dirichlet series $A_3(s,\tfrac12)$ is shown to be meromorphic for $\Re s > 23/28$ with at most simple poles at $s=1$ and $s=5/6$ and polynomial growth between them, so the Mellin integral defining the moment can be shifted to $\Re s = 23/28+\varepsilon$, and the residue at $s=5/6$ — the trace of the polar line $s+w=4/3$ through the Gauss-sum series $H_{3,1}$ — is computed in closed form and equals $C'_3$.
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On the paper's own terms, the central claim is that the boundary geometry of the target set, summarized by one tube constant $C_{\partial A}$ and one codimension $q$, enters the effective error in joint Frobenius–Artin equidistribution explicitly and uniformly. For a CST-motive $M$ with motivic Sato–Tate group $G$, a union $C$ of conjugacy classes of $\Gamma = \mathrm{Gal}(L/K)$, and a conjugacy-invariant set $A$ whose pullback to a maximal torus has boundary of upper Minkowski codimension $q$, Theorem 5.3 gives $\pi_{A,C}(x) = (|C|/|\Gamma|)\,\mu_{ST}(A)\,\mathrm{Li}(x)$ plus an error $O_G(n_K |C|^{1/2}\mu_{ST}(A)\,x^{1/2}\log(S_{L/K}(M)x))$ and a boundary term $O_G(n_K C_{\partial A} |C|^{1 - q/(2\delta_G)} |\Gamma|^{1 - q/\delta_G} x^{1 - q/(2\delta_G)} (\log x)^{2q/\delta_G - 1} [\log(S_{L/K}(M)x)]^{q/\delta_G})$, valid beyond an explicit threshold $x_0$. A further layer is uniformity over moving targets: Theorem 5.11 treats an $(\alpha,q)$-Minkowski family $A_p$ indexed by primes, replacing the main term by $\sum \mu_{ST}(A_p)$ and the boundary exponent by $\beta_G = \frac{q}{2}\min\{1/\delta_G, (1+2\alpha)/(\delta_G+1)\}$, which degrades in the Hausdorff drift rate $\alpha$. The paper further shows that quotient symmetries replace $\delta_G$ by the smaller $\delta_N$ of the quotient group and thereby improve the exponents, and it derives applications to motivic Atkin–Serre asymptotics, least-prime bounds, and Picard-rank jumps of K3 surfaces via their transcendental motives.
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The central claim is Theorem 1.1: for nonzero real $d_1,\dots,d_n$ with $|d_j|\neq|d_k|$ whenever $j<k$, any compact sets $K_j$ inside the strip $1/2<\mathrm{Re}\,s<1$ with connected complements, and any continuous non-vanishing functions $f_j$ holomorphic in the interiors of the $K_j$, the set of $\tau\in[T,2T]$ for which $\max_j \sup_{s\in K_j}|\zeta(s+i d_j\tau)-f_j(s)|<\varepsilon$ has positive lower density. The paper also proves that this statement is best possible: $d_j=0$ makes the $j$-th shift constant in $\tau$, equal parameters make two shifts identical, and opposite parameters force complex-conjugate values, so each configuration blocks simultaneous approximation of two independent functions. As a corollary, the generalized strong recurrence property -- the shifts $\zeta(s+i d_j\tau)$ staying simultaneously $\varepsilon$-close to one another on a positive-density set of $\tau$ -- holds for arbitrary nonzero real parameters, and the paper observes that a family containing both a zero and a nonzero parameter has this recurrence property exactly when the Riemann hypothesis holds.
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On its own terms, the paper's central discovery is Theorem 1.1. For a primitive character $\chi$ modulo $q$, coprime integers $m,n\le T$, and $H\in[\sqrt{T},T]$, the twisted second moment $$\$int_T^{{T+H}}$|L(\$\sigma$+it,\chi)|^2\left(\frac mn\right)^{it}dt$$ equals, for $1/2<\sigma\le1$, the integral over $[T,T+H]$ of $$\frac{\chi(m)\chi(n)}{(mn)^\$\sigma$}L(2\$\sigma$,\chi_0)+\left(\frac{qt}{2\pi}\right)^{1-2\$\sigma$}\frac{\chi(m)\chi(n)}{(mn)^{1-\$\sigma$}}L(2-2\$\sigma$,\chi_0)$$ plus an error of size $O((qT)^{1-\sigma}mn\log^2(qT))$ and a symmetric twist error of size $O(Hq^{1/2}(qT)^{-\sigma/2}((m/n)^{\sigma/2}+(m/n)^{-\sigma/2})\log^2(qT))$, with absolute implied constants. At $\sigma=1/2$ the two main terms collapse into an explicit formula: $\varphi(q)q^{-1}\chi(m)\chi(n)(mn)^{1/2}$ times $((T+H)\log(T+H)-H-T\log T)$ plus $H$ times the constant $2\gamma_E+2\sum_{p|q}\log p/(p-1)+\log q-\log(2\pi mn)$. From this asymptotic the paper derives normalized moment ratios $I_m/I_1$ (Theorems 3.1 and 3.3) and proves that the one-level density of the family $\{L(s+it,\chi)\}$ weighted by $|L(\sigma+it,\chi)|^2$ converges to $\int\phi(x)W_{U,\sigma}(x)\,dx$ with error $O(1/\log T)$ for test functions with Fourier support in $(-\sigma/(\sigma+1),\sigma/(\sigma+1))$; when the weight uses a different character $\chi_2\ne\chi_1$, the same limit is the plain unitary density $W_U(x)=1$, a stability phenomenon the paper presents as new.
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The paper's central claim is that the logarithmic decay law Keith Briggs observed numerically — $\log\Delta(n)\approx -\frac{1}{2}\log\log n$ along colossally abundant numbers, with $\Delta(n)=e^\gamma\log\log n-\sigma(n)/n$ — is a forced consequence of Ramanujan's conditional inequalities, not an empirical coincidence. Using the normalized gap $T(n)=\Delta(n)\sqrt{\log n}$, the paper recalls Ramanujan's bounds (2.4), which put the liminf and limsup of $T$ along CA numbers between $c_1\approx 1.3932$ and $c_2\approx 1.5578$; since $\Delta(n)=T(n)/\sqrt{\log n}$ and $\log T(n)=O(1)$, this yields $\log\Delta(n)=-\frac{1}{2}\log\log n+O(1)$ (equation 2.7), with a bounded remainder that is stronger than a leading-slope statement. The bounds themselves come from Ramanujan's posthumously published notes and from related work the paper cites; the paper's own contribution is the derivation of the log law, the record-value reduction of the test set, and the numerical comparison at the next order. In that comparison, $T(n)$ is measured against the oscillatory sum over zeta zeros $R(n)=e^\gamma\left(2(\sqrt{2}-1)-2\sum_{j\ge 1}\frac{\cos(t_j u)}{1/4+t_j^2}\right)$: the residual $D=R-T$ is small and slowly varying on both CA and CA2, while $Q(n)=e^{-\gamma}uD(n)$ rises from roughly $0.36$ to $1.25$ over the sampled range without settling. The paper states explicitly that the observations do not determine a limiting coefficient, and leaves the convergence of $Q$, the proposed bound $Q\le 3/2$, the equality of CA and CA2 limits, and the existence of a constant-coefficient asymptotic expansion as open questions.
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On the paper's own terms, the discovery is that Serre's weak conjecture holds for $K=\mathbb{Q}(\sqrt{13})$: for every odd prime $\ell$ and every continuous, absolutely irreducible, totally odd representation $\rho\colon \operatorname{Gal}_K \to \operatorname{GL}_2(\mathbb{F}_\ell)$ of odd conductor, there exists a Hilbert modular form $f$ over $K$ with $\rho \cong \rho_{f,\ell}$. Theorem 5.1 establishes this assuming GRH and Assumption 1, the paper's explicitly stated postulate that a Skinner-Wiles-Pan style modularity lifting theorem holds in the mixed residually reducible case (ordinary at one prime above $p$, irreducible at the other), where the literature so far covers only the two pure cases. Theorem 5.2, which the paper describes as technical and ad hoc for this particular field, then drops Assumption 1 at the price of two stated instances of it whose proofs, the paper says, will appear in a forthcoming appendix. The proof combines a new reduction-to-level-1 theorem (Section 4), valid for any real quadratic field with narrow class number one, with a base case (Theorem 2.8) specific to $\mathbb{Q}(\sqrt{13})$: a two-round discriminant squeeze shows that under GRH any level-one mod-3 representation with absolutely irreducible image is modular, because the upper bound on the root discriminant of its fixed field, $\sqrt{13}\cdot 3^{13/6} \approx 38.97$, collides with the GRH lower bound before any projective image beyond $\operatorname{PGL}_2(\mathbb{F}_9)$ can fit.
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The central discovery, stated as Theorem 3, is that the unshifted Jensen polynomials of every Dedekind zeta function are asymptotically real-rooted: for each number field $K$ there are constants $C_K>0$ and $d_0(K)\ge 2$ such that for every integer $d\ge d_0(K)$, the polynomial $J_{K,d}(X)$ is nonzero and has at least $d - C_K\sqrt{d\log d}$ distinct negative real zeros. Equivalently, the proportion of real zeros is $1 - O_K(\sqrt{\log d/d})$, and the zeros that are counted lie in an explicit interval: in the variable of $J_{K,d}$ they occur for arguments contained in $[-C_1 d^{-1}(\log d)^2,\, -c_1/\log d]$. The proof is unconditional, assuming neither GRH nor $\zeta_K(1/2)\neq 0$, and the possibility that the central value vanishes — a phenomenon the paper illustrates with a cited example — is absorbed into the 'sufficiently large $d$' formulation, since $J_{K,d}$ is identically zero only below the order of vanishing of $F_K$ at zero. The authors are explicit about the boundary of the result: a growing number of nonreal zeros is still permitted, and the estimate yields neither a zero-free region nor a zero-density estimate, so it does not by itself decide GRH.
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The paper's central claim is Theorem 1: for $0<t_1<t_2<K^{1/5}$ with $t_2-t_1>1/\log K$ and $1/2+1/\log K<\sigma<1$, the $\psi$-weighted harmonic average over even weights $k$ of the zero count $N(\sigma,t_1,t_2,\operatorname{sym}^2 f)$ is $O((t_2-t_1)K^{1-(1/19)(\sigma-1/2)}\log K)$. The route is the classical mollified-moment scheme: control zeros by showing that $L(\sigma+it,\operatorname{sym}^2 f)$ times a short Dirichlet polynomial (the mollifier) is close to $1$ on average. The new input is a twisted second moment asymptotic (Proposition 1) evaluating the harmonic average of $|L(\sigma+it,\operatorname{sym}^2 f)|^2\lambda_f(r^2)$ as a sum of three explicit main terms $S_+,S_-,S_{OD}$ plus an error. The off-diagonal part is evaluated by computing the same second moment twice, once pairing $\lambda_f(m^2)$ with $\lambda_f(r^2)$ and once pairing $\lambda_f(n^2)$ with $\lambda_f(r^2)$; a limiting-sequence lemma (Lemma 7) forces the two off-diagonal integrals to differ only by a term that vanishes, which yields a symmetry under $t\mapsto -t$ that cancels the poles at $\pm it$ and leaves the single residue $S_{OD}$ as the effective main term. From this, Proposition 2 shows the mollified product is $1$ up to a small error and that its harmonically averaged square is the natural main term plus $O(K^{1-(1/9)(\sigma-1/2)})$, which a classical zero-counting lemma (Lemma 11) converts into the count of Theorem 1.
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The paper's central claim is a double complexity classification of $\mathsf{SparseIrred}_{\mathbb{Z}}$, the problem of deciding irreducibility in $\mathbb{Z}[X]$ for polynomials presented sparsely, where the input size charges only the logarithms of exponents and coefficients. Theorem 4 states that the problem is coNP-hard under deterministic polynomial-time many-one reductions: an exact-3-CNF formula $\Phi$ is encoded as a polynomial $F = A + X^M(X^N - 1)$ whose $N$-th roots of unity correspond, via the Chinese remainder theorem, to truth assignments of $\Phi$, and a gap theorem (a specialization of a simultaneous-approximation and polynomial-lifting argument) forces $F$ to be irreducible in $\mathbb{Z}[X]$ exactly when $\Phi$ is unsatisfiable; equivalently, reducibility is NP-hard, and irreducibility is NP-hard even under one-query Turing reductions. Theorem 10 states that the problem is in $\mathsf{PP} \subseteq \mathsf{PSPACE}$: the weighted modular-root count $A_f(X) = \sum_{p \le X}(\nu_f(p) + \mu_f(p))$ detects repeated as well as distinct factors, the average $A_f(X)$ approximates $\rho(f)\pi(X)$ with an error controlled by a zero-free half-plane $\operatorname{Re} s > 31/32$ for Dedekind zeta functions and by an effective prime-ideal counting estimate, and the inequality $2A_f(X) < 3\pi(X)$ at an explicit cutoff $X = 2^{32k}$ (with $k = O(L)$) decides irreducibility. The $\mathsf{PP}$ membership is realized by a majority machine whose accepting-minus-rejecting path count equals $2(3\pi(X) - 2A_f(X))$, and the same criterion yields a deterministic streaming algorithm that runs in polynomial space.
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The paper establishes that for any L-function F of degree 1 in the Selberg class (including shifts of Dirichlet L-functions) and for several degree 2 functions (Hecke cusp forms, Maass cusp forms, Dedekind zeta of quadratic fields, products L(s,χ1)L(s,χ2) with q1q2 not a square), we have max_{T≤t≤2T} ± Z_F(t) ≫ exp(C_F √(log T / log log T)) where C_F is explicit: 1/2 for degree 1, 1/√2 for cusp forms and the listed special degree-2 cases, and √(κ(1-γ)) for general degree 2 under an exponential sum hypothesis and the Selberg normality conjecture. The proof is based on Soundararajan's resonance method with a Dirichlet polynomial resonator, and requires exponential sum estimates for the Dirichlet coefficients. The paper also obtains an unconditional result for Maass cusp forms with constant √(25/71) using the Kim-Sarnak bound.
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The central result is Theorem 1: $M(x) = C(\log(3x))^2 + O\big(\log x\, (\log\log\log x)^{1/10} / (\log\log x)^{3/10}\big)$ unconditionally, and with error $O(\log x / (\log\log x)^{1/2})$ under the quasi-Riemann hypothesis. This improves on the previous best error $O(\log x \log\log x)$ due to McShane and Rivin. The constant $C$ is explicit and equals $\operatorname{vol}(B_X)/(12\,\zeta(2)) = 2\operatorname{vol}(B_X)/\pi^2$, where $B_X$ is the McShane–Rivin unit ball for the modular torus. The proof also gives an improved error term for the number of simple closed geodesics on the modular punctured torus (Theorem 13).
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The paper's central claim is that the distinct positive ordinates $\Gamma^+ = \{\gamma_1 < \gamma_2 < \cdots\}$ of the nontrivial zeros of the Riemann zeta function are recovered as the $a \to \infty$ limit of the positive spectra of a distinguished family of self-adjoint operators, where each finite-stage operator is defined purely by local arithmetic data (the archimedean place together with the finitely many prime powers $p^m \le e^{2a}$), with no zero information as input. Under RH, Theorem 1.1 states that for the extension $D_{a,\pi}$ of the differentiation operator $i\,d/dx$ on the Hilbert space $H(A_a)$ determined by the Weil quadratic form localized to $[-a,a]$, the $n$-th positive eigenvalue satisfies $t_{a,n} \ge \gamma_n$ and decreases monotonically to $\gamma_n$ as $a \to \infty$. Theorem 1.2 upgrades this to the level of entire functions: the normalized spectral functions $W(a,\pi;z)/W(a,\pi;0)$ converge locally uniformly on $\mathbb{C}$ to the canonical product $\Delta_\Gamma(z) = \prod_{\gamma \in \Gamma^+} (1 - z^2/\gamma^2)$ over the distinct ordinates, which does not record multiplicities. The engine of the proof is Theorem 3.3, the inverse-compression identity $(D_{a,\pi})^{-1} = j_a^* D_{\pi/2}^{-1} j_a$: the inverse of the local extension is the compression to $H(A_a)$ of the inverse of a global self-adjoint operator $D_{\pi/2}$ whose spectrum is exactly $\Gamma$. Compressing this Hilbert-Schmidt inverse converges in Hilbert-Schmidt norm to the global inverse, the max-min principle turns that convergence into the monotone eigenvalue bounds, and Hadamard factorization of the even entire function $W(a,\pi;\cdot)$ with the tail control supplied by $t_{a,n} \ge \gamma_n$ yields the locally uniform entire-function limit.
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The central claim is that the normalized discriminant $\Delta_0(T) = K(T)/Y(T)^6$, built from the logarithmic derivative of the transverse product with its apparent singularities cleared, carries a compact genus-two real spectral core for the Riemann $\Xi$-function. Directed $\theta$-moment and Rouché/Krawczyk certificates isolate exactly six simple branch values — two real, two nonreal conjugate pairs — so the normalization of the double cover $y^2 = \Delta_0(T)$ over a conjugation-invariant domain is the hyperelliptic sextic $X_2: W_X^2 = P_X(T)$ of real type $(2,1,1)$: genus two, one real oval, nondividing. Theorem 3.12 certifies real parameters in an explicit box such that the sextic numerator of the van der Put–Saito $P_{\mathrm{III}}(D_6)$ scalar normal form satisfies $N_6(sT+r)/t^2 s^6 = P_X(T)$, exhibiting the Xi core as the orientation curve of an actual real $D_6$ Lax phase point. The arithmetic theorem (Theorem 5.2, Corollary 5.3) is that the calibrated zero-action lattice $\Lambda_2$ is trivial exactly when both real period ratios $u_1/u_2$ and $r_1/r_2$ are irrational; since that irrationality is not proved, the paper instead certifies the Farey windows $13749/419336 < u_1/u_2 < 20219/616667$ and $16579/8678 < -r_1/r_2 < 252355/132091$, forcing reduced denominators $q_u \ge 1{,}036{,}003$ and $q_r \ge 140{,}769$, uniformly for $|\kappa| \le 3\times10^{-10}$, with positive certified ratio velocities giving generic local nonresonance and Diophantine separation under the heat flow. Separately, a mollified reciprocal-head canopy certificate with exactly enumerated integer convolution norms, combined with the published rectangular time-barrier and finite-RH verifications, yields $\Lambda \le 433/2500 = 0.1732$ in the heat-time normalization $t = 4\kappa$ of the de Bruijn–Newman flow, with no quasi-Riemann-hypothesis assumption. The paper asserts that the heat tangent differs from the matched fixed-exponent isomonodromic direction, admitting only a local moving-exponent lift, and it explicitly disclaims any equivalence between core nonresonance and RH.
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On its own terms, the paper's central claim is Theorem 1.1: under Hypothesis 2.1 — the uniform two-over-two zeta-ratios formula (2.3) with error $O_\varepsilon(U^{1/2+\varepsilon})$ for shifts with real parts in $[3/64,5/64]$ and imaginary parts of size at most $U^{1/4}$ — one obtains $\sum_{n\le x}\Lambda(n)\Lambda(n+4)=2C_2x+o(x)$, the same asymptotic for $A_4(x)$, and $\pi_4(x)\sim 2C_2x/(\log x)^2$, where $C_2$ is the twin-prime constant and $S(4)=2C_2$ is the singular series for difference four. The quantitative engine is Theorem 1.2: for a fixed real smooth weight $W$ supported in $(1,101/100)$, the correlation sum with weight $W(n/x)^2/n$ equals $2C_2\int_0^\infty W(u)^2\,du/u$ with error $O_{W,\varepsilon}(x^{-1/16+\varepsilon})$, improved to $O_{W,\varepsilon}(x^{-3/8+\varepsilon})$ if the Riemann hypothesis is added. Two corollaries extend the reach: the logarithmically weighted prime-pair sum satisfies $B_4(x)=2C_2\log x+o(\log x)$, and the second finite difference of integrals of squared increments of the truncated Chebyshev function equals $4C_2x+o(x)$, tying the pair problem to a finite Nyman-Beurling approximant. Throughout, the zero-free strip (2.6), the zero-count bound (2.8), and the differentiated ratio mean (2.14) are derived from the hypothesis itself rather than assumed.
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The central claim is Theorem 1.1(i): under RH, the set {ζ_r(1/2+iγ): γ a non-trivial zero ordinate of ζ} is dense in C for every r≥4. The proof begins with the harmonic-product decomposition specialized to equal arguments, ζ_r(s) = (1/r!)∑_Π c(Π)ζ(s;Π) with c(Π) = (−1)^{r−ℓ}∏(|P_j|−1)!. Partitions containing a singleton block produce terms with a factor ζ(s), so at s=1/2+iγ these terms vanish because ζ(1/2+iγ)=0 under RH. The remaining partitions give a polynomial P_{1/2+iγ}(X) evaluated at X=ζ(1+2iγ). For r≥4 this polynomial has positive degree, so every complex number z has a preimage a under P_{1/2}. The auxiliary Theorem 1.3 then produces zero ordinates, with positive lower density along a suitable subsequence, for which ζ(1+2iγ) is near a and the relevant prime-angle shifts are simultaneously small; this forces ζ_r(1/2+iγ) to be near z. For r=3 the polynomial is constant and the values reduce to ζ(3/2+3iγ)/3, which is bounded by ζ(3/2)/3.
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The paper's central claim is that one mechanism reaches two classical gap constants. Theorem 1 states that $\mu \le 1/2$ unconditionally for every fixed primitive Dirichlet character $\chi$, meaning that for every $\varepsilon>0$ infinitely many consecutive normalised gaps fall below $1/2+\varepsilon$. Off-line zeros occur in pairs with identical ordinates, so infinitely many of them would already force $\mu=0$; otherwise all sufficiently high zeros lie on the critical line and the proof below applies in full. The argument assumes $\mu>1/2$ and derives a contradiction from the moments of the zero-counting logarithmic difference $V_\phi(t)$, whose imaginary part counts zeros in an interval of length $2\pi\phi/\log T$. Under the gap assumption this variable takes only values on two horizontal lines, so its weighted moments — evaluated exactly for every fixed order by contour integration against the squared modulus of a Möbius Dirichlet polynomial — must be the moments of a positive measure. Lemma 3 turns those moments into explicit densities on the two lines; for the test function $f(x)=e^{-i\pi\phi x}-e^{i\pi\phi}e^{i\pi\phi x}$ the lower-line density at zero equals $\sin(\pi\phi)\sin(2\pi\phi)/(\pi\phi\|f\|_2^2)$, which is negative for every $1/2<\phi<1$. A positive measure cannot have negative density, so $\mu\le 1/2$ follows. Theorem 2, assuming RH for zeta, keeps the same sign mechanism on a shortened interval $[0,\theta]$ and turns the contradiction into a counting argument that yields $\gg T\log T$ small spacings. Theorem 3, assuming RH for each fixed character, makes the weighted mean number of zeros in an interval less than one, which forces an empty interval; symmetric coefficients on reduced fractions $m/n$ reduce that mean to a ratio of two quadratic forms, and a $1638$-coefficient polynomial — found by a finite generalised-eigenvalue optimisation and certified by interval arithmetic — makes the ratio less than one at $\phi=3.63846723$.
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On the assumption that the Riemann hypothesis is true, the note claims three explicit statements about where primes must lie. Theorem 1: for every $x \geq 3.5\times 10^{90}$ the interval $[x, x+x^{1/2+1/40}]$ contains a prime — the 2001 Baker–Harman–Pintz theorem with its 'sufficiently large' threshold filled in. Theorem 2: for every $x \geq 1$ and every $0 < m \leq 15.831$, the interval $[x, x+x^{1/2+1/m}]$ contains a prime. Theorem 3: for every $y \geq 1$ and every $M \geq 2.29$ there is a prime between $y^M$ and $(y+1)^M$. The small-$x$ half of Theorem 2 is delegated to a computer verification of the upper Oppermann conjecture (a prime between $n^2$ and $n^2+n$) through $n = 7\times 10^{13}$, i.e. $x \leq 4.9\times 10^{27}$; the large-$x$ half is a direct comparison of interval sizes, with crossover at $x_* = \exp(-40\,W_{-1}(-5/176)) \approx 3.4828\times 10^{90}$ for $m=40$ and the exponent bound $m < \ln x_{**}/(\ln\ln x_{**} + \ln(22/25)) = 15.83179525$ at $x_{**} = 4.9\times 10^{27}$. Theorem 3 follows from Theorem 2 through the substitution $x = y^M$ with $M = 2m/(m-2)$, which rewrites the target interval as $[y^M, y^M + y^{M-1}]$, contained in $[y^M, (y+1)^M]$.
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For every fixed 0<k<1/2, under the Riemann Hypothesis, the negative moment I_{-k}(α,T) = (1/T) ∫$_T^{{2T}}$ |ζ(1/2+α+it)|^{-2k} dt satisfies I_{-k}(α,T) ≪ $α^{{-k^2}}$ for all α=o(1) with α ≫ 1/log T. This confirms Gonek's conjectured upper bound for the first time in this range. For k=1/2, a weaker bound is obtained that is sufficient to improve the known bound on the partial sums of the Möbius function to ∑_{n≤x} μ(n) ≪ √x exp(C √(log x / log log x)). For the minimum on intervals of length 1, the paper proves that for α small but not too close to 0, the minimum min_{|h|≤1} |ζ(1/2+α+i(t+h))| equals $α^{{1+o(1)}}$ with probability tending to 1.
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The central claim, Theorem 1.1, pairs a criterion with a computation. For each primitive real character $\chi_m$, attach the heat-flow family $H^L_t(z) = \int_{-\infty}^{\infty} e^{tu^2}\Phi_L(u)e^{izu}\,du$, whose zeros are all real exactly when $t \ge \Lambda_L$. If every zero of $H^L_0$ lies in the strip $|\operatorname{Im} z| \le D$, then $\Lambda_L \le \min\{D^2/2,\, \Lambda^\ast(m,\chi_m;D)\}$, where $\Lambda^\ast$ is the infimum of $t_0 + y_0^2/2$ over parameter triples whose three zero-free regions have been verified. At $D = 1$, the classical critical strip available unconditionally, the verified triples give $\Lambda_L \le 953/3200 = 0.2978125$ for all 242 real primitive characters with $3 \le m \le 400$, with individual bounds as low as 0.2278. The proof decomposes the required zero-free region into an initial rectangle at $t = 0$ (supplied by the published GRH verification [25]), a moving barrier swept out by the zero dynamics for $0 \le t \le t_0$, and an analytic right tail at the final time, and every error bound in the underlying effective Riemann–Siegel approximation is explicit and tracks the conductor $m$, uniform for $3 \le m \le 4\cdot10^5$.
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The paper's central claim, the Main Theorem (Theorem 17.1; Theorem 3.2 for cyclotomic base fields and their class-field descendants), states that for every number field $K$ and every finite-order Hecke character $\chi$ of $K$, $L_K(s,\chi) \neq 0$ for $\operatorname{Re}(s) > B_{\flat}$, where $B_{\flat} = 0.874886647039964\ldots$ is the root near $0.8749$ of $133488B^4 - 376596B^3 + 384534B^2 - 167541B + 26231 = 0$; the pole of the principal character at $s=1$ is permitted, and in particular $\zeta_K(s) \neq 0$ for $\operatorname{Re}(s) > B_{\flat}$, $s \neq 1$, with no assertion on the boundary line itself. The paper further claims that $B_{\flat}$ is the exact infimum of the refined comparison system built from the moment parameter $\kappa$ and the averaged-numerator estimate: every finite chain of admissible refined steps starting from a bound at least $B_{\flat}$ produces only boundaries at least $B_{\flat}$, yet chains starting from $7/8$ approach $B_{\flat}$ from above within any $\varepsilon$, so the limiting contradiction argument attains an endpoint no finite chain can reach. The larger boundary $B_{\infty} = B_{\mathrm{new}} = 0.874957019420098946\ldots$ is separately shown to be the infimum of the pointwise-numerator system, and the improvement to $B_{\flat}$ is attributed concretely to replacing the pointwise reflected-numerator bound by an amplified first moment. Because the analytic argument is asserted to use only the containment $\mu_{12} \subseteq F$ plus standard facts valid for every number field, the cyclotomic theorem is run over $F = K(\zeta_{12})$ and descended to arbitrary $K$ by class-field factorization; the paper stresses that the $\theta$ constructions and local calculations are imported from the preprint [Ope26b] but not that preprint's zero-free conclusion, and it states in Remark 3.1 that the accompanying machine-checked formalization covers only scalar and selected arithmetic implications, not the complete analytic argument of Part II.
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The central claim, stated as Theorems 1 and 2, is that under $(Q_\Theta)$ — every nontrivial zero of every primitive Dirichlet $L$-function satisfies $\Re\rho \le \Theta$ for a fixed $1/2 \le \Theta < 11/12$ — one has $E_+(N) \ll_\Theta N^{1/2}(\log N)^{100}$ and $E_-(N) \ll_\Theta N^{1/2}(\log N)^{100}$. Here $E_+(N)$ counts even integers $n$ with $2 < n \le N$ that are not sums of two primes, and $E_-(N)$ counts positive even $h \le N$ that are not differences of two primes. The bridge between the two theorems is the fourth-moment bound $\int_{\mathfrak{m}(X^\kappa)} |S_w(\alpha)|^4\,d\alpha \ll_{w,\Theta} X^{5/2}(\log X)^{100}$ for the smoothed prime exponential sum $S_w(\alpha) = \sum_p (\log p)\,w(p/X)\,e(p\alpha)$ on the minor arcs, with $\kappa = \kappa(\Theta) \in (4/9, 9/20)$. By additive orthogonality, the minor-arc Fourier coefficients of $S_w(\alpha)^2$ and $|S_w(\alpha)|^2$ have $\ell^2$-sums exactly equal to that fourth moment; on the major arcs the corresponding coefficients are at least a constant times $X$ for every even $n \in [X/2, X]$ and every even $h \in [2, X/2]$, so a missing representation contributes a full $X^2$ to the $\ell^2$-sum and forces the number of exceptions down to the square-root scale. Structurally, the proof isolates the principal character's contribution, then for the nonprincipal part associates each large value of the complete prime polynomial with a nearby zero of a primitive Dirichlet $L$-function, and balances the zero-density saving against the energy of the large-value set, with an exact four-stage identity capping the terminal decomposition at eight factors and fixing the logarithmic exponent.
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The paper's central claim is that the two theorems previously proved only for vertical arithmetic progressions hold verbatim for every real-rooted trigonometric polynomial, with no arithmetic-structure hypothesis on its zero set. The transport is one object: the Kurasov–Sarnak Fourier summation formula, which asserts that for every Schwartz test function $\varphi$, the zero-counting sum $\sum_{P(\gamma)=0}\varphi(\gamma)$ equals $c_0\widehat{\varphi}(0) + \sum_{a\in\mathbb{Z}_+^M} c_a\widehat{\varphi}(\ell\cdot a)$, with coefficients $c_a = -(a\cdot\ell)[z^a]\log(L(z)/L(0))$ of size $O(|a|)$. This formula replaces Poisson summation and converts every discrete average over $P$'s zeros into a continuous integral, the error being controlled by the polynomial growth of the coefficients. Proposition 5 shows the discrete mollified first and second moments of $\zeta$ agree with their continuous counterparts up to $o(T)$; combined with the second-moment bound of Balasubramanian–Conrey–Heath-Brown and Cauchy–Schwarz, that forces at least one third of $P$'s zeros to be free of zeta zeros, giving Theorem 1. For Theorem 2, a Soundararajan-type resonator satisfies the expected sample-size conditions (Lemma 10), the genuinely new step being the removal of a thin set of exceptional primes so that crystalline frequencies lying extremely close to logarithms of rationals cannot spoil the main term.
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On its own terms, the paper discovers that the local configuration of zeros around a fixed zero, encoded in the exponential sum $S_\pi(X,T,\rho'_\pi)$ with the weight $w(u)=4/(4+u^2)$, is statistically universal once the zero $\rho'_\pi$ is allowed to vary. Theorems 1.1 and 1.2 establish, under RH for $L(s,\pi)$, that the centered real parts of $S_\pi(T^{\alpha m},T,\rho')$ have even moments $\mu_{2d}(\alpha m\log T/4)^d+O((\log T)^{d-1/2})$ with $\mu_{2d}=1\cdot3\cdots(2d-1)$, exactly the moments of a Gaussian of variance $\alpha m\log T/4$, whence the standard-normal limit; universality in $\pi$ enters through the shared Rankin-Selberg second moment $\sum_{n\le X}|\Lambda_\pi(n)|^2/n=\log^2 X/2+O_\pi(1)$, which is identical for every $\pi$. Theorems 1.3 and 1.4 replace RH by the zero-density hypothesis $Z_\pi$ for the short-range regime $X=(\log T)^{\alpha m}$, where a secondary term $X^{-2}L'/L(\rho'-2,\pi)$ is shifted out and the variance scale becomes $\log\log T$; for $m=1,2$ these results are unconditional. Theorems 1.6 and 1.7 treat double sums over pairs of neighbouring zeros: the limiting law is Laplace(0,1) when $\alpha_1\ne\alpha_2$ and $\chi^2_2$ when $\alpha_1=\alpha_2$, the two regimes corresponding to whether the diagonal of the moment computation forces the prime products to match separately or allows full permutation symmetry.
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Under the Riemann Hypothesis, the liminf of (γ_{n+1} - γ_n) log γ_n / (2π) is at most 1/2.
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The central discovery is an unconditional proof that the factorials n! for n ≥ 8 are separated from all cubes and all fourth powers (and hence all even powers ≥ 4) by more than n. For cubes the proof uses cubic reciprocity and the planar Hermite bound; for fourth powers it uses a character estimate and a product comparison that avoids any external computation. For fifth powers the paper provides an unconditional finite interval (n ≤ 1100) and an unconditional large‑n argument (with an uncomputed effective threshold N₅), plus a complete proof under a specific GRH hypothesis. These results together show that any counterexample to the full conjecture must be either a square of the form n! = a²(b²−1) or an odd‑prime‑power equation n! + εp = m^ℓ with ℓ ≥ 5 and n/2 < p ≤ n, and also must have P⁺(m) > n+2.
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For any fixed derivative order ℓ≥0, the maximum over positive even fundamental discriminants d near X satisfies: at σ=1/2, at least exp((1+o(1)) sqrt{log X log_3 X / log_2 X}); at σ=1/2 + A/log_2 X, at least exp(($e^{{-A}}$+o(1)) V_X) with V_X = sqrt{log X log_3 X / log_2 X}; for fixed σ∈(1/2,1), at least exp((C_B(σ)+o(1)) (log X)^{1-σ}/(log_2 X)^σ); at σ=1, at least (D_ℓ(0)+o(1)) (log_2 X)^{ℓ+1}. The paper also proves finer relative bounds comparing derivatives with specific GCD sums or finite Euler products, including the crossover between two contributions in the mesoscopic band.
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Assuming the two black-box statements from [OAI26] (Scholia (I) and (II)), the paper establishes a lower bound $\Pi_K(x) \ge c_K \, x / (\log x)^2$ for the number of prime ideals of $K$ whose norm is prime and lie in $(x,2x)$ and for which the natural map $\mathcal{O}_K^* \to (\mathcal{O}_K/\mathfrak{p})^*$ is surjective. By Harper–Murty's criterion, this lower bound implies that $\mathcal{O}_K$ is Euclidean whenever $h_K = 1$. The technical heart is a new estimate for the weighted sum $\Theta_{F,\varphi}(x)$ (Proposition 1.6), which the author proves unconditionally for any number field $F$, and which, combined with the zero-free region, yields control over the prime ideals that split in certain ray class fields. The paper's novelty concentrates on the previously open degrees $[K:\mathbb{Q}]=2,3$.
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The central claim, stated as Theorem 7 and Proposition 9, is an unconditional estimate: N₀(T) ≥ (2 – w(ψ) + O(log L / L)) N(T), where N(T) is the number of zeros with imaginary part up to T (counted with multiplicity), N₀(T) counts simple zeros on the critical line, and w(ψ) is a ratio of integrals involving the window function ψ. Choosing ψ_MT(x) = cos(√2 x) gives w(ψ_MT) = 1/2 + (1/√2) cot(1/√2), hence N₀(T) ≥ 0.67250 N(T). The paper also obtains N_d(T) ≥ 0.83625 N(T) for distinct zeros (simple on the line, multiple on the line, or conjugate pairs).
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The paper proves that for any real quadratic function field K over F_q(T) (q odd) and any pair (x_1, x_2) in k_∞^2 not lying in the diagonal embedding of K, there exist infinitely many principal prime ideals p of O_K and elements b ∈ O_K such that |x_i - σ_i(b)/σ_i(π)| ≤ N(p)^{-2/3+ε} for i=1,2, where π is a suitably normalised generator of p. This is the exact analogue of Matomäki's classical exponent θ=1/3. The result is unconditional because the Riemann hypothesis for Hecke L-functions holds for function fields (Weil). The proof overcomes the obstacle of infinitely many units in the real quadratic case by using Hecke characters of non-trivial infinity type, whose infinity types form a finite set (parametrised by n modulo the regulator R). Consequently, the Fourier expansion of a Dirichlet character on normalised prime generators becomes an exact finite linear combination of Hecke character values, avoiding the truncation errors that occur in the number-field analogue.
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The paper's main result (Theorem 1) is that for integers $l,k$ with $(l,k)=1$ and large $X$, $$\sum_{\substack{n\leq X\\ P^-(n)\equiv l(\mod k)}} \frac{\mu(n)\omega(n)}{n} = \frac{C(k,l)}{\log X} + O\left(\frac{(\log\log X)^4}{(\log X)^2}\right),$$ where $$C(k,l) = -\frac{1}{\varphi(k)} + \sum_p \left(1_{p\equiv l(\mod k)} - \frac{1}{\varphi(k)}\right)\frac{\zeta(1,p)}{p},$$ and $\zeta(1,p)=\prod_{q\leq p}(1-1/q)^{-1}$. This improves on Tenenbaum's $O$-bound and gives an explicit main term. Under the assumption of GRH (Theorem 5), the paper shows that as $k\to\infty$, the number of reduced residue classes $l$ mod $k$ with $C(k,l)\geq 0$ is $O(\varphi(k)/\log k)$, so that $C(k,l)$ is negative for almost all classes. Thus the leading term is typically negative, consistent with the earlier result that the full series sums to zero.
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The authors prove that $\liminf_{T\to\infty} N^{s\cup 0}(T)/N(T) > 8/9$, where $N^{s\cup 0}(T)$ counts zeros of the Riemann zeta function that are simple or on the critical line (with multiplicity). The constant in the lower bound is approximately $0.88893$.
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The paper's main result (Theorem 1) states that for every fixed 0 < c ≤ 1/4 and ℓ = 2,3,4, the lower asymptotic density of ℓ consecutive c-moderate gaps (each at least 2πc/log T) is at least (ℓ+4 − (ℓ+2)C_MT)/2, where C_MT = 1/2 + (1/√2) cot(1/√2) ≈ 1.3274992963205. Concretely, the lower bounds are 0.3450014… (ℓ=2), 0.1812517… (ℓ=3), and 0.0175021… (ℓ=4). These hold unconditionally, without assuming the Riemann hypothesis or the pair correlation conjecture, answering a question raised by Gonek and Sahay.
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Sampling at Gram points is far from neutral: the full cloud $\mathcal{C}_N=\{E(g_n)\}$ and the prime cloud $\mathcal{P}_N=\{E_1(g_n)\}$ form highly structured subsets of the ambient curves traced by $E(t)$ and $E_1(t)$ as $t$ varies. In the full cloud the good/bad separation is forced by the approximate functional equation, which gives $Z(g_n)=2(-1)^n\Re E(g_n)+O(g_n^{-1/4})$; in the prime cloud goodness is not defined, yet the isolated bad points still organize into two lobes above and below the good-point cloud. Folding the two lobes onto each other reveals a principal direction along which the viscosity $\mu(g_n)=|Z'(g_n)/Z(g_n)|$ varies systematically: the smallest viscosities cluster at one extremal end, near the conjectural repulsion boundary $\mu=4$, and the isolated-bad viscosity distribution has a sharp lower edge just above that value, with an observed minimum of about 4.26178. The lobe dichotomy is itself geometric: the upper lobe corresponds to the confined zero pair lying to the right of $g_n$, the lower lobe to the pair lying to the left, and the extremal low-viscosity points correspond to confined pairs in which one zero is displaced toward the outer wall $g_{n\pm1}$ of the local Gram interval. The data further suggest the normalized asymptotic lower limit $\liminf (2\pi/\log(g_n/2\pi))\,\mu(g_n)=2$ for isolated bad confined points. All of this happens at modest height, with $N(g_n)\le 164$ so the prime layer has at most 38 terms, a caution the paper itself flags against over-reading the rank correlations. The Davenport–Heilbronn analogue exhibits neither the viscosity bound, since its isolated-bad distribution has substantial mass below $\mu=4$ with a representative value of about 1.45, nor the two-lobe splitting, indicating that both phenomena are tied to the zero-protecting dynamics of the Hardy $Z$-function rather than to the formal functional equation alone.
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The central discovery is that, under H_{7/8}, the exceptional set $E_\square(N)$ satisfies $E_\square(N) \ll_\varepsilon N^{11/20+\varepsilon}$ for every $\varepsilon>0$. The improvement comes from a new uniform estimate (Lemma 6) for weighted zero‑counts of families $C_d(T,K)$ of primitive characters obtained by twisting a fixed quadratic character $\lambda_d$ with characters of varying conductors $t$. The argument controls the fourth moment of $L(1/2+iu,\chi)$ over such families using an approximate functional equation and the large sieve, then applies zero‑detection to absorb all bands of zeros with real part at least $129/167$. The remaining bands and the minor arcs give the final exponent $11/20$.
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On the paper's own terms, the discovery is a chain of four conditional theorems. Assuming QRH 7/8 — that all nontrivial zeros $\rho$ of the Riemann zeta function satisfy $\operatorname{Re}(\rho) \le 7/8$ — the paper establishes, for all $n \in \mathbb{Z}^+$: $\pi((n+1)^8) > \pi(n^8)$, $\pi(n^8 + 8n^7) > \pi(n^8)$, and $\pi(n^8 + n^7) > \pi(n^8)$, and for all real $x > 1$: $\pi(x + x^{7/8}) > \pi(x)$. Each inequality states that at least one prime exists inside an explicit window; the narrowest integer window has relative length $1/n$, and the real window has relative length $x^{-1/8}$, so all four windows shrink to a vanishing fraction of the location as the numbers grow. The derivation is completely explicit: the hypothesis enters only through a companion preprint's two-sided estimate on the second Chebyshev function $\psi(x)$, which the paper converts into asymmetric bounds on the first Chebyshev function $\vartheta(x)$ and evaluates at the endpoints of each window, with the finite leftover ranges disposed of by direct computation. The paper states plainly that none of this stands without the hypothesis, which it describes as claimed by an unreleased AI-generated preprint collection rather than proved.
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For every integer k ≥ 1, the secondary zeta function at even and odd arguments can be written as absolutely convergent integrals: Z(2k) = ((-1)^k)/(2^k (k-1)!) ∫_0^∞ (D_k L(x))/$x^{{2k+1}}$ dx and Z(2k+1) = ((-1)^k)/(π (2k-1)!!) ∫_0^∞ (D_k L(x))/$x^{{2k+2}}$ dx, where L(x) = log(ξ(x+1/2)/ξ(1/2)) and D_k = x d/dx ∏_{j=1}^k (x d/dx − 2j). These representations replace earlier series expansions with integrals that involve only the ξ-function and its derivatives, and they unify the treatment of even and odd arguments.
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The paper's central claim is Theorem 1.1: for a Galois extension L/K with Galois group G, if ζ_L(s) has no zeros with Re(s) > 1 − δ for some fixed δ ∈ (0,1/2], then for every x > 2 and every conjugacy class C ⊆ G, the error in the Chebotarev count obeys |π_C(x,L/K) − (#C/#G) Li(x)| ≤ c_1 (#C/#G) $x^{{1−δ}}$ log($Δ_L^{{1/δ}}$ $x^{{n_L}}$), with c_1 an effective absolute constant. The new ingredient, relative to the classical Lagarias–Odlyzko treatment, is a lemma controlling the contribution of the low-lying zeros: under the fixed-width strip assumption, every zero has |ρ| ≥ δ, and this converts the sum over low-lying zeros into a bound of size log($Δ_L^{{1/δ}}$) rather than a power of log Δ_L. From this single estimate the authors deduce, still under QRH(δ), that every conjugacy class has a prime ideal of norm at most (log Δ_L)^{1/δ}(log log Δ_L)^{2/δ}, that there are class-group torsion bounds of the Ellenberg–Venkatesh shape #Cl_L[ℓ] ≪ $Δ_L^{{1/2 − 1/(2ℓ(d−1)) + ε}}$, that moments of ℓ-torsion over arbitrary families of degree-d extensions obey the Koymans–Thorner-type bound, and that the smallest generator of L has Weil height at most C_{n,δ} $Δ_L^{{1/(2n)}}$.
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Theorem 1.1: under premises (A1)-(A3) (source correspondence, finite localisation, and uniform contact reduction), the de Bruijn-Newman constant satisfies Λ ≤ 148611789321711 / $10^{15}$ = 0.148611789321711 < 0.148612. This is the smallest classical-strip bound certified in the paper, obtained by a directed comparison that starts from the classical zero strip |Im z| ≤ 1. The improvement over the previous two-source comparison is 0.000021760452379. The paper provides a fully certified rational arithmetic layer but notes that full analytic validation of the premises remains open.
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The paper establishes a general moment-to-density theorem (Theorem 1.1) for globally tempered automorphic L-functions on short strips. Applied to the Riemann zeta function using the known short-interval fourth moment (Heath-Brown, Iwaniec), it obtains N_ζ(σ; T, T+T^a) ≪ $T^{{a B(σ)+ε}}$ with B(σ)=15(1-σ)/(3+5σ) for 7/10 ≤ σ ≤ 4/5 and 2/3 ≤ a ≤ 1. This improves the earlier exponent of Ye–Zhang on that σ-range and localizes the Guth–Maynard full-interval bound to intervals of length T^a. The paper then derives explicit power savings for smoothed prime-power sums with phase exp(2π i α n^β) for fixed α≠0, 0<β<1, using the explicit formula. The minimal interval length thresholds for obtaining a power saving are 2/3 (for zeta), 3/4 (for GL(2) cusp forms twisted by a real character), and 5/6 (for self-dual GL(3) Maass cusp forms). Uniform frequency dependence for mα with 1 ≤ m ≤ K^R is also given.
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The central claim, Theorem 1.1, is that if the analytic inputs of [OAI26] hold — the family-wide $7/8$ zero-free half-plane, the marked inverse moment, the sixth-power amplification, the plain fourth moment, and the simultaneous-witness lemma — then every finite-order Hecke $L$-function over $F = \mathbb{Q}(\sqrt{-3})$, and by norm-pullback every Dirichlet $L$-function including $\zeta(s)$, has no zeros in $\operatorname{Re}(s) > B_1 = 8748867/10000000$. The proof runs the licensing chain $\beta^* \le 7/8 \Rightarrow \kappa = 3/4 \Rightarrow \beta^* \le B_0 = 69991/80000 \Rightarrow \kappa = 29991/40000 \Rightarrow \beta^* \le B_1$, where $\beta^*$ is the supremum of the real parts of the family's zeros and $\kappa$ is the plain-moment parameter; Section 2 is explicit that imported results are hypotheses of the theorem, and the ordering ensures the second $\kappa$ is used only after $B_0$ has been proved. Each stage is closed by Lemma 8.1: if a zero lay to the right of the claimed boundary, the reciprocal $1/L(s,\eta)$ would both extend holomorphically past that line, by the low and high estimates of the same compensated probe, and have a pole there, a contradiction. The boundary improves the previous value $B_L = 0.87495707\ldots$ by about $7\times 10^{-5}$, and Corollary 9.1 converts the widened half-plane into the bound $O(X^{B_1+\varepsilon})$ for smoothed Möbius sums with Hecke character weights.
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Theorem 2.1 states that for large T and t ≍ T, the imaginary part of Z(½+it) equals the real part of a sum over all nontrivial zeros ρ of A∞(−iL(ρ−½−it)), plus an error O(1/L) with L=log T. Here A∞(z)=2(z−sin z)/z² is the sine transform of the triangular weight (1−u), making it the odd counterpart of the squared sinc kernel for the real part. The proof uses a recursive relation between moments of the prime sum and a high-order Fejér identity for the real part, then telescopes the derivatives. Corollary 2.2 gives the asymptotic average of the imaginary part at zeros shifted by a microscopic distance, showing a non-zero maximum at d≈0.4147.
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The central claim is Theorem 21.1: the family supremum $\beta^*$ over real parts of nontrivial zeros of all primitive finite-order Hecke $L$-functions over $F = \mathbb{Q}(\sqrt{-3})$ satisfies $\beta^* \le B_{\mathrm{new}}$, and consequently every finite-order Hecke $L$-function over $F$, every Dirichlet $L$-function, and $\zeta$ itself is zero-free on $\mathrm{Re}\, s > B_{\mathrm{new}}$, with principal poles at $s = 1$ allowed. Numerically $7/8 - B_{\mathrm{new}} = 4.293 \times 10^{-5}$, but the improvement is structural rather than numerical: the parent $7/8$ argument fixed the parameters at $b = 1/8$, $\ell = 1/6$, while the balanced comparison boundary $B(\ell) = 11/12 - \ell/4$ shows that increasing $\ell$ lowers the boundary, subject to the nonprincipal error exponent $E(b, \ell; \delta, x)$. The new parameters are forced by the critical zero class where the character-count exponent $R^* = 2/3$ cannot be improved by changing $b$: $\delta_c = (49 - \sqrt{921})/48$ is the root of $288\delta^2 - 588\delta + 185 = 0$, $\ell_{\mathrm{new}} = (33 + 8\sqrt{921})/1653$ solves $E = 0$, and $b_{\mathrm{new}} = -4/29 + 230\sqrt{921}/26709$ solves stationarity in $\delta$; the $\sqrt{921}$ is the discriminant of that quadratic, not a free choice. At those parameters the endpoint inequality is an exact quadratic identity with equality only at the critical pair, and Proposition 25.1 proves that under the three explicit affine constraints of the same estimate system every admissible boundary must satisfy $B \ge B_{\mathrm{new}}$. The intermediate rational bound $B_r = 34999/40000$, obtained by moving $\ell$ to $1/6 + 1/10000$ with $b$ fixed, is verified by an exact sign calculation in the Bernstein basis, and the paper reports machine-checked formalizations of both bounds.
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The paper claims to have proven that the exponential sum of the Möbius function with linear phases satisfies |∑_{x<n≤x+H} μ(n)e(nα)| ≪ H (log x)^{-1/3+ε} for H ≥ $x^{{0.58+ε}}$, uniformly in α, and that the analogous sum for the von Mangoldt function satisfies |∑_{x<n≤x+H} Λ(n)e(nα)| ≪ H (log x)^{-A} for any A unless ∥qα∥ is very small for some small q. These results improve the previous admissible exponent from 3/5 to 0.58, and the method shows that any further refinement of large value estimates for Dirichlet polynomials will lower the exponent further.
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The central claim is Theorem 1.1: under GRH, for every fixed $B>2+2e^{-1/3}$ and all sufficiently large $N$, every $g\in G$ (the integers that are neither $-1$ nor a perfect square) with $|g|\in[N,2N]$ satisfies $p_g\le(\log|g|)^B$, with at most $O((\log N)^{B-1})$ exceptions. The proof runs the Hooley/Fan–Pollack $\ell$-test machinery, splitting the test primes into three ranges $I_1,I_2,I_3$; the large-prime range is handled by the new larger-sieve variant, which leaves only a set of $g$ of the claimed size because each exceptional $g$ need only fall into a positive proportion of the forbidden residue classes rather than into all of them. The same framework yields tighter conditional exponents under restrictions: $3.35$ for positive $g$ and $3.22$ for squarefree $g$, both with correspondingly smaller exceptional sets. A separate elementary argument via the Chinese remainder theorem gives the unconditional statement that almost every $g\in[M,M+N]$ has $p_g\le\Phi(N)$ for any $\Phi(N)\to\infty$.
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Theorem 1.3: assume GRH for Dirichlet L-functions with characters mod q. Then for every j at least 3, G_{q,j}(N) = (C_{q,j}/phi(q)) * G_{1,j}(N) + O_q(N^{j-1} log^2 N), where C_{q,j} = phi_{j-1}(q)/phi(q)^{j-1} is positive unless both j-1 and q are even. If correct, the average over multiples of q of the j-prime von-Mangoldt-weighted representation count is fully determined by the constant C_{q,j} and the known unweighted average G_{1,j}(N) ~ N^j/j!.
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The central claim is that the level sets of the code-theoretic quantity O(p) are governed by the near-primitive-root theory of the number 4. Precisely, for an odd prime p, O(p) equals the index [F_p^× : ⟨−1,2⟩], which is one half of ind_p(4) when ord_p(4) is odd and 0 otherwise; the underlying graph G_p is isomorphic to the square graph H_p whose cycles all have length ord_p(4). Assuming GRH, the primes with O(p)=k have natural density δ_k = A·E(k) for odd k, density 0 for k≡2 (mod 4), and density (2/3)·A·E(k) for 4|k, with A the Artin constant and E(k) the explicit Euler-product factor of the paper, and the counting function P_k(x) has main term δ_k·x/log x. Beyond primes, the paper proves P(x) = (5/12)·x/log x + O(x log log x/(log x)^{7/6}) for the number of primes admitting a tight equi-difference code, and N_0(x) = C0·x/(log x)^{7/12} for the odd integers with O(m)=0, with C0 = 0.32235..., determined numerically by truncating an Euler product at t = $10^{11}$. When O(p) is itself an odd prime, the upper bound M(p) ≤ M_e(p) + ⌊O(p)/3⌋ is attained, and such primes, still under GRH, have relative density 0.508822711238... among primes with O(p) > 2.
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The central claim is Theorem 1.1: for F(x) = ξ(1/2 + √x) = Σ λ_n x^n with λ_n = $ξ^{{(2n)}}$(1/2)/(2n)!, one has (L^m λ)_n > 0 for all m,n ≥ 0, so the sequence is strictly infinitely log-concave. The proof works with logarithmic ratios g_{m,n} = log r_{m,n}, where r_{m,n} = A²_{m,n}/(A_{m,n-1}A_{m,n+1}) and A = L^m λ; positivity of g is exactly log-concavity at iteration m. These ratios propagate by the recurrence g_{m+1,n} = 2g_{m,n} − Δ²h(g_{m,n}) with h(t) = log(1−$e^{{−t}}$). The initial ratios are estimated from an integral representation Λ(z) = H(z)/Γ(2z+1) for λ_n, using a saddle-point estimate with the Lambert W function as saddle: |H/H* − 1| ≤ 30/T on the sector |arg z| ≤ π/4, |z| ≥ 4000, which yields |g_0(z) − Q(z)| ≤ 20/n² with Q(z) = 2w/(z(1+w)), w = W(2z/π). The iterates are then controlled on successively smaller disks, and the ratio r_{τ(n),n} is shown to exceed the golden-ratio threshold (3+√5)/2 at a finite τ(n): at τ(n) = ⌈log₂n⌉ for n ≥ 12000 and at m = 14 for the finitely many n below 12000 (the latter by certified interval arithmetic). A closure lemma shows that once r ≥ (3+√5)/2 holds at one iteration, it persists forever, because (r−1)² ≥ r exactly in that range.
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The central discovery is the adjoint translation identity (Proposition 4.1): for $\tau \in \mathbb{R}$ and $\Re s > 1/2$, the operator $T_\tau = D_\tau + C Q_\tau$ satisfies $T^*_\tau k_s = \frac{\bar s + i\tau}{\bar s}\, k_{s - i\tau}$ and $\|T_\tau\| \le 1 + 2|\tau|$. Paired with the pairings $\langle h_k, k_s\rangle = -\frac{\zeta(s)}{s}(k^{1-s} - 1)$ and the kernel criterion $k_s \in \mathcal{N}^\perp \iff \zeta(s) = 0$, invariance of $\mathcal{N}$ under $T_\tau$ forces: if $\rho$ is any zero with $\Re \rho > 1/2$, then $\zeta(\rho - i n \tau) = 0$ for every $n \ge 0$, an arithmetic progression of zeros whose ordinates grow in steps of $|\tau|$, contradicting Ingham's bound $N(\sigma, T) = o(T)$. Theorem 5.1 asserts the resulting equivalence, RH holds iff $T_\tau h_k \in \mathcal{N}$ for some fixed $k \ge 2$, and supplies the quantitative gap $\mathrm{dist}(T_\tau h_k, \mathcal{N}) \ge |\zeta(\rho - i\tau)|\, |k^{1 - \rho + i\tau} - 1| / (|\rho|\, \|k_\rho\|) > 0$ whenever $\zeta(\rho - i\tau) \ne 0$ for an off-line zero $\rho$.
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The paper's central claim is a two-part theorem. Theorem 1.2 asserts that for any congruence subgroup $\Gamma$ of any indefinite quaternion algebra over $\mathbb{Q}$ and any class function $\varphi$ on the finite quotient $\tilde{\Gamma}$, the weighted count satisfies $\Psi_\Gamma(x;\varphi)=\langle\varphi,1\rangle x+O_{\varphi,\varepsilon}(x^{\eta(\theta)+\varepsilon})$ with $\eta(\theta)$ as above, whenever $\theta$ is an admissible subconvexity exponent for quadratic Dirichlet $L$-functions in the level aspect. Corollary 1.3 deduces, for the modular group and uniformly for every group in the class, that $\Psi_\Gamma(x)=x+O_\varepsilon(x^{\eta(1/6)+\varepsilon})$ with $\eta(1/6)\approx 0.69359<25/36$, which the paper describes as an unconditional improvement on the previous record. The proof obtains a new zero-density estimate for quadratic characters (Proposition 2.1) that replaces Heath-Brown's fourth-moment bound $x^{1+\varepsilon}$ by a subconvexity fourth moment $|B_\sigma|x^{4\theta+\varepsilon}$ over an exceptional set $B_\sigma$, and then feeds this through a smoothed trace-formula identity for $\Psi_\Gamma(x+u)-\Psi_\Gamma(x)$.
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On the paper's own terms, the central discovery is structural. The classical maximal-height argument for $H_\tau$ discards the interaction with all zeros except the conjugate, landing on $(y^2)' \le -2$. The paper proves the full interaction has a definite collective sign after projection: for the maximal-height zero $z = x + iy$, one has $\frac{d}{d\tau} y^2 \le -2 - 4y^2 G_\tau(z)$ with $G_\tau(z) = \sum_{\rho \ne z, \bar z} 1/((\operatorname{Re}\rho - x)^2 + 4y^2) \ge 0$. It then converts the apparently harder problem of lower-bounding this sum into certifying a single off-zero logarithmic derivative, $-\operatorname{Im}(H_\tau'/H_\tau)(x+i\eta)$, which the published effective approximation $H_\tau = B_\tau F_\tau$ can certify at high $x$. The payoff is a conditional landing: with a certified field constant $G_\tau > 1.60028669$ at the frontier row ($X = 6000000185827$, $\tau_0 = 129/800$, $y_0^2 = 87677/2500000$), a two-envelope comparison reduces the maximal nonreal height to $0.08$ by time $0.1747532546428610755\ldots$, and the classical contraction finishes the landing, giving $\Lambda \le 0.1779532546428610755\ldots$ — explicitly conditional on the imported, not-yet-peer-reviewed certificate package rather than on the new theorems.
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In the paper's own terms: with $x = e^{2t}$, $\widehat\xi(\tfrac12 + iz) = iS(z)$, where $S(z) = \int_0^\infty K(t)\sin(zt)\,dt$ and $K$ is positive, smooth, doubly exponentially decaying, with $K(0) > 0$. Two integrations by parts give the endpoint asymptotic $S(y) = K(0)/y + O(y^{-2})$, so $S$ keeps one sign on each end of the real axis and has only finitely many real zeros (Corollary 1). Because $S$ is an odd entire function of order at most one, if it had only finitely many zeros overall, Hadamard factorization would force $S(z) = P(z)e^{az+b}$, contradicting $S(y) \to 0$ as $y \to \pm\infty$ (Lemma 4). Hence $S$ has infinitely many nonreal zeros, and $\widehat\xi$ has infinitely many zeros off the critical line — Theorem 1, which refutes Yang's Conjecture 1.
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The paper establishes that the ω-inequality, σ(n)/n < e^γ Σ_{j=0}^{ω(n)} (ln ln ln n)^j / j! for n > 5040, is equivalent to Robin's inequality and hence to the Riemann hypothesis. The forward direction is immediate because the truncated sum lies strictly below the full exponential e^(ln ln ln n). The reverse direction, RH ⇒ ω-inequality, is proven by combining a conditional bound of Nicolas with a strong-tail estimate: once a minimal counterexample is known to have ω(n) ≥ 21, the truncation error is shown to lie below 6.4×$10^{-15}$ / √(ln n), well inside the 0.095/√(ln n) margin in Nicolas' bound. As a corollary, RH is equivalent to the ω-inequality holding only on colossally abundant numbers.
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The central claim is that the zero condition zeta(s)=0, combined with the Euler–Maclaurin decomposition zeta(s)=X_N(s)-Y_N(s)+R_N(s), forces the ratio Q_N(s)=|X_N(s)|^2/|Y_N(s)|^2 to tend to 1 as N tends to infinity. The paper then asserts an asymptotic rigidity theorem stating that this same ratio diverges to infinity when Re(s)>1/2, collapses to zero when Re(s)<1/2, and converges to 1 exactly when Re(s)=1/2. Since the Bridge Lemma shows a zero requires Q_N(s)->1, the only compatible real part is 1/2, so every nontrivial zero must lie on the critical line. Theorem 5.1 states this as the resolution of the Riemann Hypothesis.
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The central discovery is that the localized odd critical Weil operator, defined on $L^{2}$_odd(-a,a) with the polar rank-one term retained, is positive semi-definite for every finite support radius a. This yields non-negativity of the global Weil form on the compactly supported odd test core, which by the restricted odd Weil criterion suffices to prove the Riemann hypothesis. The argument uses an exact block recursion and Schur induction on the arithmetic endpoints a_k = (1/2) log k, with a rigorously certified base case at k = 7.
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The central discovery is that under GRH there exist cubic abelian fields of discriminant at most x whose Euler-Kronecker constant exceeds log log x + log log log x − C^+_{cubic} − ε and whose negative exceeds 2 log log x + 2 log log log x − C^-_{cubic} − ε, where C^+_{cubic} = 5.7972... and C^-_{cubic} = 9.4207... are explicit constants computed from sums over primes. The result follows from a resonance method applied to the logarithmic derivative of the cubic Dirichlet L-function at s=1, whose real part is directly related to the Euler-Kronecker constant via γ_K = γ + 2 Re(L'/L(1,χ)).
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Theorem 1.1 gives the complete list of primes p satisfying the condition ω(p − $x^{2}$) ≤ 2 for every odd x with $x^{2}$ < p. For p ≡ 7 (mod 8) the primes are 7, 23, 47, 167. For p ≡ 5 (mod 8) they are 5, 13, 29, 37, 53, 101, 173, 197, 293, 677. For p ≡ 1 (mod 8) they are 17, 41, 73, 89, 97, 113, 137, 233, 257, 353, 593, 857, 1097, 1217, 1553, 2273, 5297. For p ≡ 3 (mod 8) they are 3, 11, 19, 59, 83, 107, 227, with one possible exception that is excluded only under the Riemann Hypothesis.
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The paper's central discovery is that, without any hypothesis on the location of the zeros of ζ, the random variable P_T (a symmetrically truncated Stieltjes transform of the renormalized zero ordinates) converges in distribution to Cauchy(0,1). This is proved by showing that the underlying point measure satisfies the general integrability condition via unconditional estimates of Fujii and Selberg. Under the mild horizontal collapse assumption H(T)=o(T) (equivalent to microscopic horizontal collapse of off-critical zeros), the normalized logarithmic derivative A_T on the critical line also converges to Cauchy(0,1), with an explicit comparison error. For function fields, purity and a weighted cotangent identity give the exact Cauchy law for each nonconstant cohomological factor, and Poincaré duality yields explicit bounds for the full zeta function, leading to Cauchy limits for smooth hypersurfaces of increasing degree.
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The paper proves that at least 7% of the non-trivial zeros of the Riemann zeta function lie on the critical line; more precisely, κ₀ > 0.0700162. This is achieved by replacing Selberg's single squared mollifier with a sum of three squared mollifiers whose coefficients are chosen to solve a convex semidefinite programme on the logarithmic profile space. The arithmetic mean value at the heart of the method is reduced to a closed-form constant C[q₃] ≈ 0.6567752, and the proportion follows as κ₀ ≥ 2θ/(e c²γ) with θ → 1/4, c = 2 and γ = C[q₃].
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Theorem 1.1: Assume the Generalised Riemann Hypothesis. If $A\subseteq\{1,\dots,N\}$ contains no two elements differing by $p+1$ for any prime $p$, then $|A|\ll N^{79/80+o(1)}$. The proof uses the Fourier transform identity $0 = \int_0^1 |\widehat{1_A}|^2(\alpha)\widehat{f}(\alpha)\,d\alpha$ with a carefully constructed function $f$ involving the von Mangoldt function, a fourth divisor function, and a sieve weight. By decomposing the integral into major and minor arcs, bounding the minor-arc contribution via Vinogradov's estimate, and using GRH to control the major-arc Fourier closeness, the authors obtain a positive lower bound for the contribution near $\alpha=0$ and a negative upper bound away from zero, forcing $|A|\ll N^{1-\delta+o(1)}$. Optimising the parameters $T=Q=N^{1/48}$, $R=N^{1/8}$ yields $\delta=1/80-o(1)$.
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The Riemann hypothesis holds if and only if the Ingham function is a function of good variation (equivalently, a regular arithmetic function) with regularity index 1/2. This is Theorem 3.9.
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Theorem 1.1 states that for every fixed 0 < θ < 1, we have liminf_{T→∞} N^s_0(T,T^θ)/N(T,T^θ) ≥ c(θ) = 2 − θ/2 − (1/√2)cot(θ/√2) and liminf_{T→∞} N_d(T,T^θ)/N(T,T^θ) ≥ d(θ) = 3/2 − θ/4 − 1/(2√2)cot(θ/√2), where N^s_0 counts simple zeros on the critical line and N_d counts distinct nontrivial zeros. These are the first explicit lower bounds for these proportions that depend on the interval exponent θ.
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The central claim is Theorem 1: for every $\varepsilon>0$ and every sufficiently large integer $q\not\equiv2\bmod4$, at least $3/8-\varepsilon$ of the primitive Dirichlet characters modulo $q$ satisfy $L(1/2,\chi)\ne0$, and the same bound holds within each nonempty parity class $\chi(-1)=(-1)^j$. The point is uniformity: previously $3/8-o(1)$ had been attained only for prime conductors, and the best bound valid for all conductors was $7/19-o(1)$. The refinements are graded by the powerful part $P(q)=\prod_{\ell^\nu\|q,\ \nu\ge2}\ell^\nu$ of the modulus. Squarefree $q$, and more generally $P(q)\le q^{1/2}$, give $5/13-\varepsilon$; prime squares give $5/13-\varepsilon$ outside a sparse exceptional set of primes, surviving the adjunction of a fixed coprime factor; and moduli whose powerful part contains a divisor of size $q^{1/3+o(1)}$, in particular $q=rp^2$ with $p^{4-\eta}\le r\le p^{4+\eta}$, give the largest constant $2/5-\varepsilon$. Density-wise, all but $O(X^{3/4})$ of the moduli up to $X$ satisfy the stronger bound.
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The paper discovers that the contrast pencil—formed by compressing the shifted arithmetic Weil matrix to the zero-mean contrast space—produces a Hermitian definite generalized eigenproblem whose eigenvalues are real and, in the dimension-matched zero-side case, exactly equal the signed zero ordinates. The rank-two displacement identity of the Weil matrix ensures that the differentiation form is Hermitian, enabling this stable coordinate-free formulation. The interpolation null vector of the zero-side matrix also serves as a Rayleigh trial vector for the full arithmetic matrix, proving that the least eigenvalue vanishes under RH.
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The central discovery is that for every nontrivial zeta zero ordinate gamma_k, the real number L_k = 1/4 + $gamma_k^{2}$ admits an exact nearest-integer decomposition L_k = N_k + delta_k, where N_k = floor(L_k + 1/2) and delta_k = B̃_1($gamma_k^{2}$ + 3/4) with B̃_1(x) = {x} - 1/2 being the periodic first Bernoulli function. Consequently, the residual delta_k is not a rounding error but an exact periodic Bernoulli readout. This identity is derived from the Bernoulli-level geometry and is independent of the truth of the Riemann Hypothesis.
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The central discovery is that for each fixed x>1 and any θ<Θ (where Θ is the supremum of real parts of non-trivial zeros, called Riemann’s constant), the partial sum ΣR_T(x) = ∑_{0<|Im(ρ)|≤T} m_ρ ERi(ρ log x) is not O(T^θ). Hence limsup_{T→∞}|ΣR_T(x)|=∞, so the series diverges. This implies that the expression ∑_ρ R(x^ρ) in the conventional ordering is not a well-defined complex number and the identity π₀(x)=R(x)−∑_ρ R(x^ρ) cannot hold with that summation. The argument also extends to show the divergence of the contribution of the trivial zeros.
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Theorem 1.1 states that, assuming GRH, for sufficiently large $X$ there exist primes $p$ with $X<p\leq 2X$ and primitive cubic (resp. quartic) characters $\chi$ mod $p$ such that $|L(1/2,\chi)|$ is at least the exponential quantity given above. The constant $\sqrt{2}/\pi$ for cubic and $\sqrt{3/2}/\pi$ for quartic arise from the exponents $4/5$ and $7/8$ in the bound of Proposition 2.4, together with an optimisation of the resonator parameters $A$ and $B$.
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The paper’s central result is Theorem 1.2: assuming the strong Riemann hypothesis, for any ε > 0 and sufficiently large x, the sum over square‑full numbers n ≤ x equals ζ(3/2)/ζ(3) $x^{{1/2}}$ + ζ(2/3)/ζ(2) $x^{{1/3}}$ + Σ_{ρ = 1/12 + iγ/6, |γ| < c_1 $x^{{1/2}}$, ζ(6ρ)=0} c_{γ/6} $x^{{1/12 + iγ/6}}$ + O_ε($x^{{10ε}}$). The coefficients c_{γ/6} are effective complex constants determined by the residue of ζ(2s)ζ(3s)/ζ(6s)·x^s/s at the pole s = 1/12 + iγ/6. The presence of the zero sum is the ‘extra main term’ announced in the title. The error O_ε($x^{{10ε}}$) is essentially a power of log x, representing a far smaller remainder than the earlier conditional exponent θ = 0.1406… or any unconditional bound.
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The central discovery of the paper is Theorem 1.1, which estabilshes an unconditional quantitative relation between a single large value of S(x,t) and the local density of zeta zeros near the line Re(s)=1. The theorem says that if |S(x,t)|=x/N with N small relative to log x, then for any L up to half log x, a small disk centered near 1+it (radius proportional to L logT/(log x)^2) contains at least L/360 zeros of ζ(s). This is the first such inverse theorem for zeta sums, modeloed on the analogous result for Dirichlet character sums but with the novel feature that the pole of ζ introduces a residue term in the Fourier transform, which is shown to be negligible in the chosen parameter range.
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The core claim is the pair of theorems linking the sign of $f$ to the zeros of $\zeta$. Theorem 1.1 proves that eventual non-negativity of $f(x)=-\sum_{n\le x}\lambda(n)\frac{\log n}{\sqrt{n}}\log\frac{x}{n}$ implies RH: the Mellin transform of $f$ is $F'(s)/(s-\frac12)^2$ with $F(s)=\zeta(2s)/\zeta(s)$, Lemma 2.4 (Landau's theorem) forces a non-negative function's abscissa of convergence to be a real singularity, and the quotient has no real singularity in $(\frac12,\infty)$ — hence the transform is analytic in $\Re s>\frac12$, which rules out any zero $\rho$ of $\zeta$ with $\Re\rho>\frac12$, because such a zero would be a pole of $F'$. Theorem 1.2 is the conditional converse: assuming RH, the Simple Zero Conjecture, and the condition H3 (that $\sum_{\gamma>0}\frac{|\zeta(1+2i\gamma)|}{\gamma^2|\zeta'(\frac12+i\gamma)|}<\infty$), the paper derives the explicit formula $f(x)=P(\log x)+(\log x)\Phi(\log x)-\Psi(\log x)+O_\varepsilon(x^{-1+\varepsilon})$, where $P(L)=-\frac{L^3}{12\zeta(\frac12)}+\alpha L+\beta$ is an explicit cubic polynomial and $\Phi,\Psi$ are bounded functions built from the zeta zeros. Since $\zeta(\frac12)<0$, the leading coefficient is positive, and the corollary states that $f(x)\sim\frac{(\log x)^3}{12|\zeta(\frac12)|}$ as $x\to\infty$, so $f$ tends to $+\infty$ and is eventually positive.
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The core claim is that the arithmetic spectrum of the Euler product, not the nontrivial zeros, carries the extensive thermodynamics of a black hole. The partition function $Z_{\mathrm{ar}}(\beta)=\prod_p(1-e^{-\beta\ln p})^{-1}=\zeta(\beta)$ describes free bosonic modes with energies $\ln p$; its linear entropy $S\simeq E/T_H$ with $T_H=1$ matches the linear-in-area entropy if the arithmetic cutoff is $E_c=T_H A/4G$, and the simple pole of $\zeta$ at $\beta=1$ gives $S(A)=A/4G+\ln(A/4G)+\text{const}$, fixing the logarithmic coefficient at $\alpha=1$. Feeding this entropy into the entropy–geometry correspondence $f(r)=1-4\pi M/S'(r)$ yields $f(r)=1-2GMr/(r^2+\ell_\zeta^2)$ with $\ell_\zeta^2=\alpha G/\pi$, a value independently forced by exact validity of $dM=T\,dS$. This metric has Reissner–Nordström horizon structure without charge, a positive-energy anisotropic source satisfying the null energy condition, a curvature singularity softened from $r^{-6}$ to $r^{-2}$, and an extremal remnant at $M_*\simeq0.56\,m_P$. The second-quantized zero gas, by contrast, has entropy $\sim\sqrt{E\ln E}$, far too slow to be extensive; zeros appear only as exponentially suppressed log-periodic oscillations in the area, whose envelope is $e^{-3A/16G}$ exactly when all zeros lie on the critical line.
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The paper's central claim is that the logarithmically averaged two-point Chowla correlation can be controlled uniformly over growing shift ranges with a genuine power-of-log saving. Concretely, Theorem 1.1 asserts that for every fixed $0<\eta<1$ and $C>0$, there is $c_\eta>0$ such that for all sufficiently large $x$ one can remove a set $\mathcal E_{x,\eta}\subseteq[1,\exp((\log x)^\eta)]$ of size $\ll_{\eta,C}\exp((\log x)^\eta)(\log x)^{-C}$, containing at most one prime, and for every remaining $h$, $$\max_{h\le\exp((\log x)^\eta),\,h\notin\mathcal E_{x,\eta}}\left|\sum_{n\le x}\frac{\$\lambda$(n)\$\lambda$(n+h)}{n}\right|\ll_\eta(\log x)^{1-c_\eta}.$$ The exceptional set is generated by a possible Landau--Page real character and is empty when no such character exists, so under the generalized Riemann hypothesis the bound holds for every shift in the range. The paper also proves an absolute-saving version for every fixed polylogarithmic range, all-moment shift-window bounds with a maximal supremum over terminal points, and removal of the $x/H$ factor for windows of length $H\ge x^\theta$ with $\theta>1/3$.
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On the paper's own terms, the discovery is a fourth sum variant of the almost linear formula, built on the Gram sequence rather than on the zeros of $Z(t)=e^{i\vartheta(t)}\zeta(1/2+it)$. For each Gram point $t_n$ in the interval $(T,\varphi_1^{-1}(T))$, the signal $A_2(n)=[t_n,t_{n+1}]\times[0,Z^2(\tilde t(n))]$ has area $|A_2(n)|=Z^2(\tilde t(n))(t_{n+1}-t_n)$, where $Z^2(\tilde t(n))$ is the average of $|\zeta(1/2+it)|^2$ over the Gram interval. Theorem 1 gives $\sum_{T<t_n<\varphi_1^{-1}(T)}|A_2(n)|\sim(1-c)T$, and because the Gram spacings satisfy $t_{n+1}-t_n\sim2\pi/\log t_n$, the number of rectangles is $N(T)\sim(1-c)T/(2\pi)$, so the quotient yields the canonical formula $\frac1N\sum|A_2(n)|\sim2\pi$. This mean-value statement is unconditional, whereas the analogous statement for zero-based rectangles $A_1(n)$ is, the paper says, only available under the Mertens hypothesis. The later theorems are obtained by subdividing each rectangle into $sq$ congruent pieces, selecting $l$ of them, and substituting $T=x/(1-c)\tau$ or $T=\frac{x^m+y^m}{z^m}\frac{\tau}{1-c}$, producing limits $x$, $x/(sq)$, $lx/(sq)$, and the corresponding $\zeta$-conditions for Fermat–Wiles.
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The paper claims that Eureka, a task-conditioned Meta-Agent, compiles long-horizon tasks into dynamic obligation graphs and forms specialized Macro-Agents, completing 170/170 recursive tasks with 3,948 acceptance certificates and no observed uncertified acceptance. In the mathematics track, it advances a positivity certificate candidate for Suzuki's localized Weil quadratic form to 0 < a ≤ 69/200 = 0.345, reaching approximately 99.55% of the first-prime threshold (log 2)/2. The authors explicitly state this is not a proof of the Riemann Hypothesis and not a record for proportion of zeros on the critical line.
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The paper's central claim is a count rather than an existence: the number $a_{\omega,N}$ of even newforms of level $N$ whose $L$-function has at least one normalized zero in $(-\omega,\omega)$ obeys the lower bounds of Theorems 1.15 and 1.18. Theorem 1.15 says that if $\omega$ satisfies $-\bigl(\hat{\varphi}(0)+\tfrac12\int \hat{\varphi}(y)\,dy\bigr) < S(1,\sigma;\varphi_\omega)$, then along a sequence of prime levels $a_{\omega,N} \gg N^{7/8}\log N$. Theorem 1.18 says that if for some odd $n\ge 3$ the strict inequality $(-\mu)^n < S(n,a;\varphi_\omega)$ holds, with $\mu$ the limiting mean of the one-level density, then $a_{\omega,N}/|F_N|$ stays bounded away from zero: a positive proportion of the family carries a zero in the window. From this the paper derives that some exact integer $r\ge 1$ occurs as the number of zeros in the window on a positive proportion of forms, that forms with at least $r$ zeros have density at most $C_1/r^{2n}$, and that a positive proportion of forms have between one and $r_0-1$ zeros. The unifying mechanism is that the sign-changing test function $\varphi_\omega$ makes the odd centered-moment excess detectable only through forms with zeros inside the window, while a uniformly bounded even moment of the zero count $R_f(\omega)$ prevents that excess from being concentrated on a negligible set of forms.
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The central discovery is that the Dyson class survives the nonlinear inverse-spectral map. For each unfolded spectrum $\{\varepsilon_n\}$, the dressing construction produces a deformation $f^{(s)}(x)$ of $H_0=-\frac{d^2}{dx^2}+\frac{x^2}{4}$; projecting onto the oscillator basis gives $F^{(s)}_{mn}=\langle m|f^{(s)}|n\rangle$, and the shell weights $W^{(s)}_d$ collect the matrix-element weight at fixed $d=|m-n|$. The moments $M^{(s)}_p=\frac{1}{N_b}\sum_{m,n}|m-n|^p |F^{(s)}_{mn}|^2$ then act as the diagnostic: for even $p=2q$, $M^{(s)}_{2q}$ is a repeated-commutator norm with $H_0$. Calibrated on Gaussian $\beta$-ensembles with $\beta=1,2,4$ for GOE, GUE, and GSE, the normalized ratios $R^{(s/\mathrm{GUE})}_p$ vary smoothly with $\beta$ and separate the three Dyson classes; applied unchanged to the Riemann zeros, they place the reconstructed operators in the GUE sector, with deviations that decrease with height and sit in the low-$d$ shells.
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The discovery, on the paper's own terms, is that two lower bounds for power sums can serve as the sole engine for zero-free regions: the first power-sum inequality detects the presence of an exceptional zero through the logarithmic derivative of a shifted Rankin–Selberg product, while a real-part power-sum bound, in combination with a zero-repulsion lemma, controls how close any other zero can come to that exceptional zero. The interaction of these two bounds forces the exceptional zero to lie at distance at least a fixed power of the conductor from 1, producing the bound |L(σ,π×π′)| ≥ c $C_π^{{-ε}}$ throughout σ ≥ 1 − c $C_π^{{-ε}}$. Standard analytic properties of Rankin–Selberg L-functions are used only as input, and the classical Siegel–Tatuzawa bound for Dirichlet L-functions is recovered as a special case.
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Theorem 1.1 establishes that for q ≡ 2 mod 3, with K = k_1 + ... + k_m and shifts satisfying alpha_j << 1/g, the mixed moment S_g($\theta$^(m), k^(m)) = sum over chi in C_g of the product |L(e(theta_j)/$q^{{1/2+alpha_j}}$, chi)|^{2k_j} is bounded by |C_g| $g^{{k_1^2+...+k_m^2}}$ times the product over i<j of (min{1/|theta_i - theta_j|, g})^{2k_i k_j}, provided g > exp($4K^{2}$). This is the first upper bound of the conjectured order for shifted cubic L-function moments in the function-field setting, and it holds unconditionally because Weil's Riemann hypothesis locates all zeros on the circle |u| = $q^{{-1/2}}$. The proof adapts Soundararajan's moment method and Harper's multiscale dyadic decomposition. A direct corollary bounds moments of derivatives: sum over chi in C_g of |$L^{{(ell)}}$($q^{{-1/2}}$, chi)|^{2k} << |C_g| $g^{{k^2+2k ell}}$.
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The discovery is the complete algebraic and dynamical classification of the contour Hankel matrices $H_m(\Gamma;\phi_\Gamma)$, whose entries are the contour moments $\mu_k=(2\pi i)^{-1}\int_\Gamma \phi_\Gamma(t)^k (\Xi'/\Xi)(t)\,dt$. By the residue representation, these moments are power sums over the coordinate images of the enclosed zeros, so $H_m$ factorizes as a Vandermonde–weight–Vandermonde product. For conjugation-compatible contour-coordinate pairs, such as real-centered circles with the normalized coordinate $\phi_{x,\rho}(t)=(t-x)/\rho$, the matrix is real symmetric, and Theorem 3.4 gives its exact inertia: for $m$ at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs. Corollary 3.7 then states that the Riemann hypothesis holds if and only if $H_m(\Gamma_\rho(x);\phi_{x,\rho})$ is positive semidefinite for every admissible real center and radius. The same structure governs motion: between crossings the matrix satisfies a Lyapunov-type congruence flow, while crossing events inject signed rank-one or rank-two indefinite jumps that can be isolated by removing the known coordinate drift.
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The central discovery is that the weak interlacing relation between adjacent quotient indices is governed by a structural theorem for Jacobi spectral multipliers. For $0<\nu<2$ and $n\ge1$, define $h_{n,\nu}=\min\{1,\sqrt{1/4+e_{n,\nu}}\}$ with $e_{n,\nu}=n(n+1)(2-\nu)/(2n+\nu)$. If $F$ is a real even polynomial or entire function of order at most one with $Z(F)\subseteq S_{h_{n,\nu}}$, then every member of the endpoint pencil $M_{F,\nu}(y^n(y+t))$ lies in the one-defect class $E_{n+1}$, so $M_{F,\nu}(y^n)\preceq M_{F,\nu}(y^{n+1})$. The specializations $\nu=3/2$ and $\nu=1/2$ recover the even and odd quotient families, and the strip widths $\sqrt{15/28}$ and $1$ are sharp where asserted. For nonpolynomial even $E$ with zeros in the same strip and with simple zeros of both images in $(0,4)$, the interlacing is strict, and this is what the arithmetic applications use. The paper also determines the exact optimal strip for the first nontrivial pencil $n=1$: $h^{\mathrm{opt}}_{1,1/2}=1.065615\ldots$ and $h^{\mathrm{opt}}_{1,3/2}=\sqrt{15/28}$.
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The central claim is Theorem 1.1: for an absolute constant K > 0, the condition $n^{3}$ $log^{2}$(n+2) ≥ K $d^{5}$ forces all d zeros of $J^{{d,n}}$ to be real, distinct, and negative. The proof constructs a comparison polynomial with explicit simple positive roots, matches the first five normalized coefficients exactly, and then shows that the remaining coefficient multiplier is a small holomorphic perturbation; the wedge condition is precisely what makes this perturbation small. Theorem 1.2 then identifies the empirical distribution of the centered and scaled zeros of the actual Jensen polynomial with Wigner's semicircle law, uniformly along every sequence of pairs (n,d) staying in the wedge.
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The paper's central claim is Theorem 1: assume RH and that there is an exceptional sequence of Siegel zeros $(\beta_j,\chi_j,q_j,E_j)$ with $q_j\to\infty$ and $E=\inf_j E_j$ sufficiently large; then $\mu=\liminf_{n\to\infty}(\gamma_{n+1}-\gamma_n)\log(\gamma_n)/(2\pi)<0.4733$. The proof chooses the resonator $R(t)=\sum_{n\le L}\chi(n)G(n)n^{-1/2-it}$, where $\chi$ is the exceptional quadratic character, with $T=q^{7/3+\delta}$ and $L=T q^{1/2-\delta}=q^{17/6}$; a classical proposition about exceptional characters gives $\chi(p)=-1$ on almost all primes, which makes the diagonal part of the numerator negative and large. The off-diagonal terms are split into three ranges of the prime-variable $k$, handled respectively by bounds for shifted character correlations $\chi(m)\chi(km+r)$, by a second-moment estimate for shifted prime sums, and by a new variable-separation argument using generalized Jacobi sums, with the zero-density estimate (Proposition 2, exponent $7/3$) controlling the largest $k$ range. Combining the asymptotics yields a numerical inequality at $c=0.473275$ with the chosen polynomial weight $G_0$, forcing $I_1>I_0$ and hence a gap of normalized size below $c$. The paper also derives Corollary 2: under the same assumptions, an asymptotic distribution with normalized gaps in $\frac12\mathbb{Z}_{\ge1}+o(1)$ cannot exist.
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The central discovery is that the one-level density of the family $\{L(s,E_d)\}$ for $d$ odd fourth-power-free obeys, on average, the formula $D_{\mathcal F^*}(\phi,w,D)=\hat\phi(0)+\frac12\int_{\mathbb R}\hat\phi(u)\,du+O(1/\log D)$, where the Fourier support of $\phi$ is contained in $(-3/5,3/5)$ under GRH and in $(-1,1)$ under a quartic Patterson conjecture. This matches the expected Katz–Sarnak symmetry for these supports. From this, the authors derive the average analytic rank bounds of $13/6$ (under GRH) and $3/2$ (under the conjecture), and consequently positive proportions of twists with minimal analytic rank consistent with parity.
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Under RH for $\zeta(s)$ and $\beta(s)=L(s,\chi_{-4})$, the paper proves that for fixed $k\ge 2$ the signed count $D_k(x)=\sum_{n\le x,\Omega(n)=k}\chi(P^+(n))$ satisfies $$D_k(x)=-\frac{$x^{{1-1/(2k)}}$}{(\log x)^k}\left\{B_k+2\Re\sum_{\gamma>0}m_\rho C_k(\rho)$x^{{i\gamma/k}}$\right\}+O_k\left(\frac{$x^{{1-1/(2k)}}$}{(\log x)^{k+1}}\right),$$ where the sum over the zeros $\rho=\frac12+i\gamma$ of $\beta(s)$ is absolutely and uniformly convergent. The positive constant $B_k$ comes from the prime-square part of the Euler product and is the deterministic part of the bias. The paper then shows that even if all zero oscillations point in the most unfavorable direction, their total absolute size is strictly smaller than $B_k$; hence the brace stays positive and the outside minus sign makes $D_k(x)<0$ for every sufficiently large $x$.
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The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ for $\ell=1,\dots,5$. Theorems 3 and 4 give matching bounds for $\Upsilon(x)=\sum_{p\le x}(\log p)/p$ and $\tilde\psi(x)=\sum_{n\le x}\Lambda(n)/n$, with $A_\Upsilon(2)=A_{\tilde\psi}(2)=9.2203$. From these bounds the paper derives two-sided inequalities for the Mertens products $\prod_{p\le x}(1-1/p)
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The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n for which any r with ℓ(n)=r must carry more than 2^k distinct algebraic roots, forcing log r ≥ k log 4 − O(k/log k). Theorem 1.2 sharpens this into a discriminant statement: for any Δ one can build an odd square-free n with log n ≤ C log Δ (log log Δ)² (and under GRH log n ≤ C log Δ log log Δ) such t
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Under the Deep Riemann Hypothesis, for any fixed spectral scale T the difference between the mollified prime-power sums for 1 and a (mod N), normalized by log x/√x, equals C_N log L(1, χ_{1,a}) + O((log x)/√x). The leading growth and the principal-character noise cancel identically because the virtual character χ_{1,a} has coefficient 1 - χ_0(a) = 0, leaving only non-principal L-series special values. Hence the bias between any two reduced residue classes is asymptotically a fixed, computable constant; for N=8 the paper derives 7 > 3 > 5 > 1 (mod 8), and by Remark 1.6 the class -1 (mod N) is universally dominant.
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The paper establishes two new results. First, Proposition 7.5: for R = X^ϑ with 0 < ϑ < 4/9, the smoothed Goldbach count r_φ(N) satisfies Σ_{N≤X} |r_φ(N) − N S(N) − M(N;R) − Z(N;R)|² ≪ (X^{3−ϑ} + X^{13/5})(log X)^5, where M and Z collect, explicitly, the contribution of every zero of every Dirichlet L-function of conductor at most R. Second, Theorem 8.2: if for some A > 5/2 and δ ∈ (0,1) the sparse Hardy–Littlewood bounds δS(N)N ≤ r_2(N) ≤ (2−δ)S(N)N hold for all but at most X^{3/5} even multiples N of r̃ in [X/2,X], with X = r̃^A, then no primitive real character χ̃ mod r̃ has a real zero β̃ > 1 − c/log X. The proof of the second result runs by contradiction: if such a zero existed, the Deu
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The central object is the Dirichlet series F_ℓ(s) = ∑ φ_ℓ(n)n^{-s} = ζ(s−1)/ζ(ℓ(s−1)+1). Its nontrivial poles occur at s = 1+(ρ−1)/ℓ for each nontrivial zero ρ of ζ, so the location of zeta zeros is encoded in the poles of F_ℓ. The paper's key analytic step is to write Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + E_ℓ(x) and form the Mellin-type integral G(s)=∫_1^∞ E_ℓ(x)x^{-s−1}dx, which equals F_ℓ(s)/s − (1/(2ζ(ℓ+1)))/(s−2). If E_ℓ(x)=O(x^{1−1/(2ℓ)+ε}) for all ε>0, then G(s) is analytic in the half-plane Re(s)>1−1/(2ℓ), so F_ℓ(s) can have no poles there. But every zero with Re(ρ)>1/2 would produce exactly such a pole; hence no such zero can exist, and the functional equation forces all zeros onto the critical l
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Theorem 1.1 states that, assuming GRH, for sufficiently large X, max_{X<|q|≤2X} |L(1/2, χ_{8q})| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a resonator R_q = Σ_{m∈M} χ_{8q}(m), with M a set of N = X^{1/4-δ} square-free integers chosen to maximize the GCD sum Σ_{m,n∈M} √((m,n)/[m,n]). The core of the argument is the ratio S2/S1: S1 counts the diagonal terms m=n via a conditional character-sum estimate (Lemma 2.2), and S2 is bounded below using the same estimate plus the observation that the smooth weight ω is close to 1 when [m,n]/(m,n) ≤ X^ε. The GCD-sum bound of Lemma 2.3 gives the exponential factor, and choosing δ → 0 yields the constant 1.