claims depot shelf
The Riemann Hypothesis
Formal claims (Lean)
Stated claims
-
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.
-
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)
-
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
-
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.
-
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
-
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
-
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.
-
For any fixed c>0, the Riemann hypothesis is equivalent to the assertion that the local moments M_{q,c}(N) of the normalized Möbius polynomial P_N, sampled uniformly on the arc of radius c/N, satisfy M_{q,c}(N)=O(N^η) for every η>0 and every finite q≥1 (and also to the weaker version that only requires the bound along an unbounded set of exponents for each η). In short, subpolynomial growth of arbitrarily high finite local moments recovers the Mertens bound.
-
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 normalized logarithms are asymptotically independent standard Gaussians under μ_F. The weight F is chosen so that F(χ)=0 if any of the four central values vanishes, and F≈1 for typical χ; this removes the obstruction that log|L| is undefined at zeros. The direct corollary is that for any fixed c>0, ≫_c q ch
-
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 obtain |f₂(p) − p/4| ≫ √p log log p for infinitely many primes p, refuting Sun's conjectured O(√p) error. They further show that, assuming GRH, the upper bound O_m(√p log log p) holds for every even monomial x^m, so the log log p factor is essentially the right order in the quadratic case.
-
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]. The same limit has the integral representation M_Φ(y,δ)=∫_0^∞ Φ(x) M(y/x) dx with an explicitly displayed kernel M(x), so the murmured signal is a convolution of the test function with a fixed arithmetic density. The boundary behaviours — the limit is 0 as y→0⁺ and −Φ̃(0)/(3ζ_K(2)) as y→∞ — identi
-
The central claim is that the zero set of D(s)=Σ ζ(2n)n^{-s} is completely and unconditionally described. Using the Lipschitz summation formula termwise in the polylogarithm decomposition, the paper establishes an exact functional equation D(s)=Γ(1−s)Z(1−s) for σ<0, where the dual series Z(w)=Σ_{ω∈Ω} ω^{-w} runs over the set Ω={log k² + 2πiℓ : k∈ℕ, ℓ∈ℤ} \ {0} with a mandatory grouping of terms; the k=1 column is precisely Riemann's functional equation. Interference between the two smallest frequencies, ω♭=2 log 2 from the entire part E and ±ω♯=±2πi from ζ, controls every zero of large modulus in the left half-plane. The paper proves that D is zero-free for σ≥σ0=1.500127440..., that D has a r
-
The paper's central claim is Theorem 1.1: under GRH, for large X, max_{X<|d|≤2X, d∈F} |L(1/2, χ_d)| ≥ exp((1+o(1))√(log X log_3 X / log_2 X)). The proof constructs a set M of squarefree integers with near-maximal GCD sums and defines a resonator R_d = Σ_{n∈M} χ_d(n). Expanding the weighted first and second moments, S_1 and S_2, and applying a GRH-conditional mean-value theorem (Lemma 2.2), the ratio S_2/S_1 is shown to be at least the GCD sum of M. With the optimal GCD sum from Lemma 2.3, this yields the stated lower bound.
-
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 estimate D_{σ,γ}(T)≪(log γ(T))^{-σ} for every shift in the class F'. The border case γ(t)=log t is excluded: the key Fourier integral does not converge, and the method forces the Dirichlet polynomial length to be constant.
-
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 T=x/(1−c)τ yields a ζ-functional whose value at every fixed x>0 is x, and for Fermat rationals x=(x^m+y^m)/z^m the value is never 1. On the Riemann hypothesis, the unimodality of Z² between zeros lets the Bonnet mean-value theorem replace each zero-gap area by the area of a rectangle B(n)=Z²(t0(n))
-
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 all-degree bound |f^{(d)}(k)/d!|≤3·40^d k^{1−d} for f=log a; the model sequence q_k^{s(s−1)/2} has an exact LDL^T factorization whose whitened dilation group R_α=L^{−1}diag(q^{αi})L has generator norm at most 3/2√(rτ); and the weighted Banach algebra gives ∥h∥_A≤0.1310721. The true block therefore di
-
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 (Hypothesis 1) that the same bound persists for T as small as x^ε, and uses that extension—via a dyadic decomposition of the explicit formula for θ(x,χ_□)—to prove n(q) ≪ (log q)^{1+ε} (Theorem 3). For primes in arithmetic progressions, the analogous Hypothesis 2 yields, under GRH, a Chebyshev-type error ψ(x
-
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 9.38 then states that boundedness of E_rho3 (and hence of F) is equivalent to the combined expression being O(1). Along the way the paper proves numerous exact cancellations: the principal residue character is absent from the edge and bulk kernels, the transposition defect cancels identically when
-
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 and logarithmic-derivative weights of ζ. The paper also proves an unconditional implication: if Φ(e^{-x})=O(x^{-1/2}) near 0, then its Mellin transform G(s) is holomorphic for Re(s)>1/2, and the identity theorem applied to ζ(s)G(s)−Γ(s) rules out zeros with Re(s)>1/2; the functional equation then rule
-
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 condition holds. Transporting the theorem through the completed functional equation of a primitive newform proves that every derivative period polynomial has all zeros simple and on the unit circle, for arbitrary level, nebentypus and derivative order.
-
Guth and Maynard prove a new large-values estimate for Dirichlet polynomials that yields the zero-density bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} uniformly for 1/2 ≤ σ ≤ 1; this is the first improvement on Ingham's 1940 estimate throughout the range σ ≤ 3/4 and, when combined with earlier results, gives A(σ) < 30/13, which in turn implies the prime-number theorem in short intervals of length x^{17/30} and almost all intervals of length x^{2/15}.
-
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.
-
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.
-
For every real even Galerkin vector v the cutoff-free truncated Weil matrix evaluates the sum of an explicitly constructed band-limited test function g_v over the nontrivial zeros of zeta with multiplicity. Independently, the omitted archimedean tail past any cutoff T larger than the Galerkin band is a strictly totally positive Cauchy–Stieltjes increment, so the finite-T eigenvalues sandwich the true eigenvalues within an explicit budget B_T that behaves like (2N+1) ho log T /(\pi^{2} T).
-
We formulate the notion of a Euclidean system of ray classes and prove that every such system generates the corresponding ray class group. Assuming GRH, if K is a totally real Galois number field of degree n≥3 and p is an odd rational prime that does not split completely in K, then for every N>0 every generating set of the ray class group Cl_K^{(p)^N} with modulus (p)^N is a Euclidean system.
-
Modifying the work of Klagsbrun, Mazur, and Rubin, the authors prove that under the extended Riemann hypothesis the Selmer ranks in the twist families are distributed so as to give bounds on the probability that an elliptic curve gains rank in p-cyclic extensions, bounds on the average size of C(L) for superelliptic curves C, and analogous probability bounds for hyperelliptic curves in quadratic extensions, all with extensions ordered by the product of ramified primes.
-
As Q tends to infinity, at least 1/9 of the zeros of L(s, Π₀ × χ) lie on the critical line, where Π₀ is a cuspidal automorphic representation of PGL(3,A_Q) and χ runs over primitive Dirichlet characters of conductor ≤ Q. The result is unconditional for self-dual Π₀ and holds under a mild condition otherwise. For representations of PGL(2) the statements are fully unconditional and give a stronger proportion. The proof relies on a new power-saving asymptotic for the mean square of L(s, Π₀ × χ) times an arbitrary Dirichlet polynomial valid for T up to Q^{1/3-ε}.
-
An overview is provided of quantum models of the Riemann zeta function that link the Hilbert-Polya conjecture to the Riemann hypothesis, together with new results on p-adic quantum computing and on quantum entanglement realized through lattice spin models and algebraic models.
-
A universal estimate is proved showing that the contribution of large primes to the singular product decays like the reciprocal of the logarithm, regardless of the structure of the system. For linear systems with trivial Galois group superfast convergence is obtained. For nonlinear systems a coefficient is defined that is expressed via the average over the Galois group; in the abelian case and under the Riemann Hypothesis for Dirichlet L-functions a more precise error estimate is obtained. Mixed systems are also considered.
-
The authors prove that the Diophantine equation P_s(n) = t^m for m > 2 has only the solutions listed in Theorems 1, 2 and 3 when s belongs to the families s = 2k+4 (k=4,6 or prime 5≤k≤97) and s = k+4 (k=9,15 or prime 3≤k≤97). Although a fully unconditional proof is not obtained for all possible solutions, the authors expect no further solutions on the basis of the generalized Riemann hypothesis and the weak effective abc conjecture.
-
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 unshifted bound ∫_0^T |ζ_K(1/2+it)|^{2k} dt ≪ T (log T)^{[K:Q]k^2}, the conjecturally sharp order. The theorem thus closes the upper side of the moment problem for Dedekind zeta functions of finite Galois extensions, conditionally on GRH.
-
Assuming the Generalised Riemann Hypothesis for L(s,χ) and that the non-trivial zeros ρ=½+iγ of L(s,χ) are simple, the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2} and ∑_{0<γ≤T} |L(2ρ,χ²)/L'(ρ,χ)|² are at least a positive constant times β/(1+β) times their conjectured leading asymptotics, where β=log T/log qT, uniformly in the conductor q.
-
The Berry-Keating operator H_BK can be analyzed from a purely Hilbertian standpoint that relies on dilation operators and the Mellin transform, and from a distributional standpoint that employs ladder operators, generalized eigenstates of H_BK, and generalized coherent states; the two standpoints are offered as complementary routes toward clarifying the operator’s still-unsettled connection to the Riemann hypothesis.
-
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.
-
Every zero produces opposite vertical curvatures on the two horizontal sides of the pole of the logarithmic derivative, so a naive two-sided vertical concavity criterion for Ξ'/Ξ cannot prove the Riemann Hypothesis. A finite spectral averaging framework replaces this obstruction by proving cancellation at the critical line, positivity of the off-critical paired contribution on the left under a concrete low-frequency kernel condition, a conditional zero-density consequence, and a precise statement of the additional localization hypotheses needed to imply the Riemann Hypothesis.
-
A natural map exists between projective varieties V(F1) and Cuntz-Krieger algebras O_A. The K-theory of O_A calculates the Frobenius action and the cardinality of V(F1^r). The zeta function of V(F1) satisfies all of Weil's conjectures except an analog of the Riemann hypothesis. The crossed product structure of O_A establishes a morphism Spec(Z) to Spec(F1) isomorphic to a point.
-
We prove a function field analogue of Duke's equidistribution theorem for CM points, in the setting of Drinfeld--Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger's formula expresses through central values of automorphic L-functions, bounded in Lindelöf-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.
-
Assuming the Riemann Hypothesis and that all non-trivial zeros are simple, the sum over those zeros of the squared modulus of a fixed ratio of zeta functions is at least half the value Ng conjectured for the same sum.
-
Conditional on GRH for L(s, χ), for every K > 0 there exist C_K and T_0 such that (1/T) ∫_T^{2T} |X_χ(t)|^{2k} dt ≤ (C_K k L_T)^k for all T ≥ T_0 and 1 ≤ k ≤ K L_T, where L_T = log log(qT), q is fixed squarefree odd ≥ 3, χ is primitive non-principal, and X_χ(t) = Im log L(1/2 + it, χ). The proof adapts Selberg's pointwise formula by splitting into three prime-power Dirichlet polynomials and applies Soundararajan's mean-value lemma to their moments.
-
The screw function provides a representation of the Weil quadratic form by continuous functions that is compatible with the constructions of Yoshida, Bombieri, and Connes-Consani. This representation leads to the conjecture that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function arises as the limit, as a tends to infinity, of self-adjoint operators coming from nonlocal realizations of the first-order differential operator on the finite interval [-a,a]. All statements hold without assuming the Riemann Hypothesis.
-
Under the generalized Riemann hypothesis, the order of magnitude of E|∑_{n≤x} h(n)λ(n)|^{2q} is determined up to factors of size e^{O(q^2)}, for all real x, q with 1 ≤ q ≤ c log x / log log x and c > 0 a small constant, where λ(n) are the Fourier coefficients of a fixed modular form and h(n) is a Steinhaus or Rademacher random multiplicative function.
-
The question of whether a zero could occur away from the critical line becomes equivalent to whether the solutions of a pair of Laplace's equations with well-defined boundary conditions in some semi-infinite strip can possess zero contour lines that intersect within that strip.
-
We show that one can always find such integers with n1,n2≤q^{2+o(1)}, unless the sign of h strongly pretends to be a real Dirichlet character modulo q. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function λ this improves on a recent result of Ford and Radziwiłł and matches, up to q^{o(1)} factors, what was previously known conditionally under the generalized Riemann hypothesis.
-
For weak Lorentz ideals associated with regularly varying functions, Dixmier traces admit a direct construction in terms of eigenvalue sequences. This yields a complete spectral characterization of measurable operators, answering a question of Connes. Weyl operators, those with precise asymptotic limits for rescaled eigenvalue sequences, form a closed subset of the ideal that remains stable under compact perturbations. Every Weyl operator is strongly measurable, so spectral measurability implies strong measurability. The theory applies to operators linked to the Riemann hypothesis (assuming RH), Schrödinger operators with anisotropic potentials, Dirichlet Laplacians on infinite-volume domain
-
Extending the Bettin-Gonek framework, the authors prove that suitable bounds on the mollified second moments of GL_m automorphic L-functions, holding for mollifiers of arbitrary polynomial length, imply that these L-functions have no zeros in corresponding regions inside the critical strip. They further show that the θ=∞ conjecture for a family of such L-functions implies a quasi-Riemann hypothesis for the family.
-
The paper shows unconditionally that L(n) = O(n log log n / log n) and B(n) ≥ c log n for large n, nearly confirming Sloane's conjecture that L(n) = O(n / log n). Under the Riemann Hypothesis, for large n, B(n) = O(n^{3/4} (log n)^{1/2}) and L(n) ≥ c' n^{1/4} (log n)^{-1/2}.
-
Let N be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of N as the sum of three primes, one of which is smaller than a given U. By inserting the currently best zero-density estimate for Dirichlet L-functions, we may unconditionally take U = N^{4/49} exp(log^{2/3 + ε} N) for any ε > 0. If we assume the Generalized Riemann Hypothesis instead, we may take U = log^{4 + ε} N.
-
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 sieve constant C1=8 in the upper bound for counts of prime pairs with the recent C1=6.7814, extended to counts with fixed difference and fixed residue class.
-
We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.
-
For any elliptic curve E over Q, in the family of its quadratic twists E_d by discriminants d, the coefficients in the Taylor expansion of the L-function L(E_d, s) around s=1 are nonzero whenever d is sufficiently large, assuming the generalized Riemann hypothesis. Unconditionally, the number of such nonvanishing coefficients in the family admits a general positive lower bound derived from moment results on the central values of derivatives of quadratic twists of modular L-functions.
-
It is shown that any number of distinct primitive GL(1) and GL(2) L-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such L-functions. The joint distribution of GL(m) L-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet L-functions and generally improve upon the work of Mahatab, Pańkowski and Vatwani on the class of L-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, 1
-
We prove that every sufficiently large integer n can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every n > 24 and prove two results to support this claim. First, we prove the result holds unconditionally for every odd n > 24. Second, assuming the Generalised Riemann Hypothesis for Dirichlet L-functions, we prove the result holds for every n > 24. We also discuss the obstruction which prohibits us from proving the result unconditionally for every n > 24.
-
Assuming the Riemann Hypothesis, we show that for k>0, (1/T) meas{t in [T,2T]: |zeta(1/2 + i t)| > (log T)^k} is at most C_k (log T)^{-k^2} / sqrt(log log T), where C_k = exp(e^{c k}) for an absolute constant c>0. This implies that the 2k-moments of |zeta| on the critical line are bounded above by C_k (log T)^{k^2}. The argument proceeds via a recursive scheme that iteratively reduces the large-deviation problem to moment estimates.
-
By summing the identities involving the functions g(n), |g(n)|, and C_Ω(n) under the probabilistic independence assumptions for Ω(n) and μ²(n), the limiting asymptotic growth of |M(x)| / √x is recovered without relying on the Riemann hypothesis.
-
The central discovery is a direct equivalence between the Riemann Hypothesis and the Salem integral equation, meaning the hypothesis is true if and only if the equation satisfies the stated property that encodes the location of the zeta zeros.
-
The duality measure K equals one over prime density plus one over zero density remains stable across scales and converges after normalization to the universal infrared fixed point K_IR equals 4 with critical exponent b approximately 0.51. This scaling law is derived from a variational information action and is viewed as the renormalization-group flow of a conserved information current from an ultraviolet fixed point of 11 down to the infrared value of 4. The generator kappa with kappa squared equals ijk equals negative one imposes, via the exchange symmetry between prime and zero information, the fixed point where both informations equal 2, thereby encoding the critical line Re(s) equals 1/2
-
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 poles activate at times t_n ∝ ρ_n³ and condense into an infinite-time essential singularity.
-
Assuming the Generalized Riemann Hypothesis, the product ∏_{j=1}^4 L(1/2 + it, χ χ_j) is nonzero for a positive proportion of Dirichlet characters χ modulo q, where q is a sufficiently large prime depending on the fixed even characters χ_j mod D_j (pairwise coprime and square-free) and on t.
-
The central claim is that for twist-inequivalent non-CM normalized newforms f and g with integer Fourier coefficients, the largest prime factor P of a_f(p) + a_g(p) satisfies P(a_f(p) + a_g(p)) > (log p)^{1/14} (log log p)^{3/7-ε} for almost all primes p and any ε > 0. Beyond primes, Brun's sieve yields the same phenomenon for a set of positive integers with natural density one. Under the generalized Riemann hypothesis the absolute value |a_f(p) + a_g(p)| grows exponentially with p. If a_f(p) + a_g(p) remains small on a positive-density subset of primes then f and g are twist-equivalent by a quadratic character.
-
We introduce the resonance-correlation method to study small gaps between consecutive zeros of the Riemann zeta-function. Our method is based on a synthesis of Montgomery's pair correlation approach and the Montgomery-Odlyzko method. As an application, we break the persistent practical barrier around 0.515 and prove μ < 0.50895 under the Riemann Hypothesis.
-
For a thin subgroup H of the full character group modulo q the L-functions L(s, χ) with χ ∈ H satisfy explicit upper bounds on |L(σ, χ)| throughout 1/2 ≤ Re(s) ≤ 1; these bounds are obtained from new zero-density estimates averaged over H that in turn rest on a mean-value theorem for character sums extending Heath-Brown's work. The same mean-value theorem supplies an unconditional bound on the minimal gap between primitive roots modulo q.
-
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 average over multiples of q is of smaller order than the unrestricted average, illustrating that k-th powers are not uniformly distributed among residue classes modulo q.
-
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})}.
-
We study generalized Skewes numbers as the first locations where two comparable prime counting functions change sign. For the race between quadratic residues and quadratic nonresidues modulo q, we construct sequences of highly composite moduli q such that these Skewes numbers grow very rapidly, disproving unconditionally a conjecture of Fiorilli. Assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes numbers. Our method uses a quantitative Kronecker-Weyl theorem in the 1-Wasserstein metric to obtain explicit rates for convergence to the limiting distributions.
-
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 conjectures that their information measures sum to zero.
-
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)} (log x)^{1/2+ε}. The same machinery, with additional linear-independence assumptions, gives a large-deviation estimate for ν_k([V,∞)) and a conjectured precise oscillation with iterated logarithms. The argument's engine is an explicit formula expressing the partial sum as a sum over zeros of the match
-
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)=∫_{-∞}^{∞} w^{2z} e^{w^2} dt = π/Γ(1/2−z) = cos(πz)Γ(z+1/2), valid for all z∈C. The scaling identity G(z,α)=α^{-(z+1/2)}G(z) turns each Dirichlet term (n+a)^{-s} into a ratio of G-integrals, and summing the geometric series produces ζ(s,a)=R(s-1/2,a)/G(s-1/2) with R(z)=∫ w^{2z} e^{w^2}/(1−e^{w^2}) dt, a meromorphic representation valid for all s∈C\N without analytic continuation or strip restrictions. The paper also derives the alternating eta version, a symmetric integral X(τ), and explicit
-
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^2)( log log t − log 2 + 1/2 + 5/(8 log log t) + 10.7084/(log log t)^2 ). These are the sharpest explicit conditional bounds currently known on the 1-line, with optimized constants in all lower-order terms.
-
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 derives the bound from the Mellin transform of E^AN_{σ1}, which is ζ(s)ζ(s−1)/(s(1−s)) plus pole terms, by shifting the contour to the critical line and applying RH bounds for ζ(s).
-
The central claim is that GRH for all non-principal Dirichlet characters modulo q is sufficient for Chebyshev's bias in the square-root weighted counting function. For distinct invertible residue classes a and b, and every ε>0, the natural density of the set of x for which |π1/2(x;q,a)−π1/2(x;q,b)+M(q;a,b) log log x − C| ≤ (log log x)^(3+ε)/log x exists and equals 1. The constant M(q;a,b) encodes the bias: when no character has a zero at s=1/2, it reduces to (r(a)−r(b))/(2φ(q)), so the class containing more square roots among reduced residues has fewer weighted primes on a set of density one. The paper also proves a density-one statement for the partial Euler products that appear in the deep
-
The central claim is that the first moment of quadratic Hecke L-functions in the Gaussian field admits an asymptotic expansion with a secondary main term whose size is the cube root of the leading term. Specifically, for a smooth compactly supported weight Φ, the sum over square-free primary d of L(1/2, χ_{(1+i)^5 d}) Φ(N(d)/X) equals X Q1(log X) + X^{1/3} Q2(log X) + O_ε(X^{1/4+ε}) under GRH, where Q1 and Q2 are linear polynomials whose coefficients are absolute constants. For general s with 1/3 < Re(s) < 1, the analogous formula contains four main terms arising from four poles of the double Dirichlet series A(s,w), with the error expressed in terms of β, the supremum of the real parts of z
-
The central discovery is that the entire algebra B — polynomials with complex coefficients in arbitrary derivatives of cuspidal automorphic L-functions — has a universal zero-counting behavior controlled by three simple invariants: the rank-weighted degree, the conductor-weighted degree (formed from the arithmetic conductors of the components), and the first index of a nonzero Dirichlet coefficient. Theorem 1.1 gives the explicit asymptotic N_F(0,T)=α₁T log T+α₂T+O_F(log T), with α₁=(1/2π)deg_rk(F) and α₂=(1/2π)(deg_cond(F)−deg_rk(F)log(2πe)−log n_F), for all F whose dominant monomials (index set J) have coefficients summing to a nonzero number. Theorem 1.2 asserts that, whenever each compon
-
Assuming the generalized Riemann hypothesis, the one-level density of zeros for the family of Γ₁(q) L-functions agrees with the Katz-Sarnak unitary prediction for test functions whose Fourier transforms have support in (-8/3, 8/3). This confirms the random matrix model for this unitary family and implies that at least 62.5% of the forms in the family are non-vanishing at s=1/2.
-
For a multiplicative f with f(p^k)=ε_k, writing z=ε_1 and w=ε_2-ε_1(ε_1+1)/2, the paper proves (under RH, SZC, and a convergence condition) that A_f^exp(x) - Δ_1(x) = Δ_{1/2}(x) + Σ_ρ Δ_ρ(x) + O(x^a), where Δ_1, Δ_{1/2}, Δ_ρ are explicit Laplace integrals with Watson-type asymptotics. Consequently the normalized difference has limit c_{1/2}(z,w) when Re(z+w)>0, bounded logarithmic-Cesàro oscillation to c_{1/2} when Re(z+w)=0, and unbounded growth when Re(z+w)<0.
-
By deriving explicit constants in Selberg's result on the average of the argument function of L(s, χ) for non-principal characters χ modulo prime q, the authors prove that for all sufficiently large such q the smallest height of a non-trivial zero in the family is at most 1075 · (2π / log q). They further establish a positive lower bound on the proportion of characters for which the first zero lies within a specified multiple of the average spacing.
-
At integer arguments the asymptotic expansions of the finite spectral sums L_n(s, χ) terminate exactly due to their structural polynomiality, producing exact identities for special values of Dirichlet L-functions. These identities include new infinite families of relations and recover prior results by a different mechanism. The special values admit interpretations as counts related to rooted spanning forests on cyclic graphs. For the zeros, the framework reformulates the generalized Riemann hypothesis for odd primitive characters via an asymptotic functional equation linking the completed discrete functions ξ_n at s and 1-s.
-
A Redheffer-type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. More general Redheffer-type matrices are considered and intriguing number-theoretic examples are illustrated. Several asymptotic results are discussed and a new expression related to the Riemann hypothesis is presented.
-
Under hypothesis (H), Theorem 5 states that for every s in the critical strip with Re(s) ≠ 1/2, the pair (μ(s), μ(1−s̄)) cannot equal (1,1). Here μ is defined by an absolutely convergent integral. When η(t) = {t}, the fractional part, the paper claims that (H) is satisfied by an estimate whose proof is not written out in the text. Combining the theorem with the integral representations (9) that link μ to ζ, the paper concludes that the non-trivial zeros of ζ lie on the critical line, thereby answering the dynamical conjecture it references.
-
The central claim, in the abstract and proved in the body as Eq. (2), is the closed formula A_w(q) = sum_{\emptyset\ne S\subseteq{0,...,n}} (q-1)^{|S|-1} gcd(k_S,q-1), with k_S = gcd{w_i: i in S}, for the number of F_q^*-orbits on nonzero F_q-representatives for the weighted scaling action. The proof uses Burnside's lemma over this action, stratifying F_q^{n+1}\{0} by coordinate support and using that the number of λ with λ^{k_S}=1 is gcd(k_S,q-1). The body further proves normalization relations, counts for singular and weak loci, rationality of the zeta function via multiplicative orders, a Riemann-hypothesis magnitude bound, and failure of the standard functional equation. The abstract's s
-
The central discovery is Theorem 1, an explicit version of a classical mean-value result with all constants spelled out. For any A(x) of polynomial growth with Mellin transform F(s)/G(s) regular in left half-planes and satisfying |F(s)| ≤ c_F max(1,|t|^{B_F})e^σ and |G(s)| ≤ c_G max(1,|t|^{B_G})e^{|σ|}, a simple zero ρ0=β0+iγ0 of G with F(ρ0)≠0 yields (1/Y)∫_1^Y |A(x)|dx ≥ an explicit expression in terms of Y, ρ0, and the growth constants. For the Mertens function, with F(s)=s−1 and G(s)=s(s−1)ζ(s), the constants are c_A=1, C=1, c_F=√2, B_F=1, c_G=13.38, B_G=7/2, and choosing the first zeta zero gives D_M(Y) ≥ 7×10^{-9}√Y for Y≥10^{20}. The proof uses contour shifts, a residue at the zero, a
-
Theorem 3.1 asserts that for any Dirichlet character χ, with log q = o(log T), the proportion κ(χ) of non-trivial zeros of L(s, χ) on the critical line exceeds 0.4172 for large T, and the proportion κ*(χ) of zeros both on the line and simple exceeds 0.4074. This generalizes the 1989 two-fifths result for zeta, and the report also proves Levinson's theorem: at least one-third of zeta's non-trivial zeros lie on the line. Both rest on one mechanism: Theorem 3.2, an asymptotic for the mollified second moment with error T^(1−ε0) (ε0 > 0) valid for mollifier length θ = 4/7 − ε under a special coefficient shape, converted by Levinson's inequality into a zero count. The longer mollifier — beyond Lev
-
Theorem 1.1: |N_K(T) − (T/π) log(d_K (T/(2πe))^{n_K}) − 1.919| ≤ 0.194(log d_K + n_K log T) + 5.543 n_K + 0.462 for T ≥ 1, and Corollary 1.2: |N(T) − (T/(2π)) log(T/(2πe))| ≤ 0.097 log T + 3.962 for T ≥ 1. If the proof were correct, these are the best explicit zero-counting error terms currently available.
-
The paper's central claim is that the martingale decomposition of S_n by largest prime factor—writing S_n = Σ_{p≤n} M_p(n) with M_p(n) = Σ_{ℓ≤n, P(ℓ)=p} θ_ℓ—allows a uniform, RH-conditional control of the whole sum. The key estimate is Lemma 3.2: under the Riemann hypothesis, Σ_p ||M_p(n)||²_{L∞} ≤ C M_n² n^{4/3+o(1)}. From this bound, Burkholder's inequality yields Theorem 2.6 and the Azuma–Hoeffding inequality yields Theorem 2.9. The weak-convergence results are independent in spirit: a Taylor expansion of the characteristic function isolates the fresh primes in (n/2, n] and forces a_n ≫ √(n/log n) for any convergent normalization, while a second argument, conditional on Assumption 2.10, s
-
The paper's core discovery is Theorem 8.2: RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120. The proof runs through a chain of unconditional estimates—one-visit and parent-window lemmas limiting how often a trajectory hits a window, macro-step alignment showing that L ≍ log X composite steps contract the scale from X to X^(3/4), and a frequency-netting lemma that controls the sum over Riemann-zero contributions using an explicit smoothed formula with cubic-log truncation and a grid-based large sieve. Iterating the contraction inequalities yields the unconditional bound E(X) ≪ X^(1/2) log X (Corollary 6.6), and the paper then argues that
-
Under the Riemann Hypothesis and mild conditions on shifts α_j, Conjecture 1 states that the discrete shifted moment Σ_{0<γ≤T} ζ(1/2+iγ+α_1)···ζ(1/2+iγ+α_k) equals the δ-derivative at δ=0 of an integral over t of one zero-swap term Z_{α_1,...,α_k,δ} plus k one-swap terms (t/2π)^{−α_j−δ} Z_{...,−δ,...,−α_j}, together with (T/2π)log(T/2π), and an error O(T^{1/2+ε}). The Z functions are products of zeta values at 1+shift divided by ζ(1+α_j), times an Euler-product arithmetic factor A. Expanding the shifts and differentiating recovers Conjecture 3: the leading asymptotic for a product of n_j-th derivatives is (−1)^{Σ n_j + k} n_1!···n_k!/(Σ n_j +1)! times (T/2π)(log(T/2π))^{Σ n_j+1}. The k=2 cas
-
On the paper's own terms, the central claim is Conjecture 1: assuming the Riemann Hypothesis, for Re(k)>−3, the average of ζ′(1/2+iγ)^k over zeros γ with 0<γ≤T is asymptotically (log(T/2π))^k / Γ(k+2). The derivation shows that a Haar-averaged unitary characteristic polynomial has exact complex derivative moments e^{iπk/2} Γ(N+k+1) / (N! Γ(k+2)), asymptotic to e^{iπk/2} N^k / Γ(k+2); the same leading constant, without the factor i^k once θ-differentiation is translated to t-differentiation, is obtained for the zero factor in the hybrid model. The prime factor separately has average 1 over the zeros. The branch is not obtained by continuous variation of log ζ′(s), but by the product represent
-
The central claim is Theorem 5.1: for the smoothed measures nu_T and mu_Omega with Omega = kappa T, the KL-regularized prime–zero transport cost satisfies OT_eta(T) << T log^2 T unconditionally. The author's route is to normalize the cost kernel to eta(t)(1 - cos gamma t), average over zero frequencies with a Fejér kernel so the cosine term becomes an average of cos(xi t), calibrate the probe's zero-frequency mass so the main density cancels, and then bound the residual by an L1-controlled smoothed explicit formula. On the paper's telling, the log^2 T factor comes from gamma/zero bookkeeping in the explicit formula, while the Paley–Wiener mass of the probe supplies the scale T. No use of RH
-
The central result is Theorem 1.1: for B a division quaternion algebra, along prime levels q, Conjecture A holds under GRH with an effective rate (log q)^{-1/4+ε} for every ε>0. Concretely, the pushforward measures (ι_q)_* |F_q|² μ_q converge weakly to μ_1 ⊗ μ_1, and the proof exhibits explicit polynomial control in the spectral parameters of the test functions. The mechanism is spectral: Weyl sums for the pair (f1, f2) are expanded over newforms φ on Y_q, each term is converted by the Watson–Ichino formula into a ratio of triple-product L-functions, and the problem becomes a fractional moment estimate for L(1/2, F⊗F⊗φ) L(1/2, f1⊗f2⊗φ). The authors establish that this fractional moment decay
-
Section 3 states: 'the Riemann hypothesis would be true should no bounded measurable function f satisfy (f(x) star e^{sigma x} k(e^x))(z)=0, other than the trivial case f identical to 0.' Theorem 1.2 further claims that condition (ii), F bar-I_{sigma,m}(t)=0 for t<-m, and condition (iii), Ff(t)=0 for t<0, are linked through the Riemann hypothesis. If true, this would give a Salem-type reformulation of RH in terms of Fourier support and Hilbert transforms.
-
The paper states: 'In all the cases we study, so long as the largest factor of N is at least a fixed power smaller than N, the lower-order terms agree to this degree of precision with those from the case when the level is prime. On the other hand, the lower-order terms differ when the smallest prime is at most a given size.' If correct, the 1- and 2-level densities of holomorphic cusp newforms have family-dependent lower-order terms up to O(1/log^4 R), with explicit formulas when one prime factor q1 is fixed, for example SA'(F) = -2 log(q1)/log(R) * hat_phi(0)/(q1^2 - 1) - log(q1)/log(R) * hat_phi''(0)/(q1^2 - 1) + O(1/log^5 R) in Theorem 7.1.
-
In the authors' terms, the paper demonstrates a decorrelation phenomenon for global Bessel periods of SO(5)×SO(2) averaged over imaginary quadratic fields, for symmetric cubes of algebraic regular Hecke eigenforms on GL(2). Concretely, if two such eigenforms are taken, the associated Bessel periods, summed over the same family of imaginary quadratic fields, no longer retain a mutual correlation: the average of the product approaches the product of the averages. The proof is conditional on the Generalized Riemann Hypothesis.
-
The paper establishes that the vanishing problem for 3-pointed genus-zero Gromov–Witten invariants on partial flag varieties has an Arthur–Merlin protocol: assuming GRH, a probabilistic polynomial-time verifier can be convinced that the invariant is zero or nonzero with a short proof. This places the problem in AM and hence in the second level of the polynomial hierarchy. The route is constructive: for each such invariant, the paper builds an explicit finite system of polynomial equations obtained by translating the defining equations, and proves an extension of the Parametric Hilbert Nullstellensatz that reduces the invariant's vanishing to the solvability of that system. The reduction is u
-
The central claim is that, under the generalized Riemann hypothesis, the shifted moment obtained by averaging L(1/2 + α1, χ) ... L(1/2 + αk, χ) over the family of primitive cubic or quartic Dirichlet characters of conductor q satisfies an explicit upper bound, uniformly in the shifts αj, with the main growth being a power of log q. The proof uses the analytic continuation, functional equation, and a GRH-conditional approximate functional equation for these higher-order L-functions to reduce the moment to short Dirichlet polynomials, then bounds the averaged sums. Once the shifted-moment bound is established, the paper derives bounds for the moments of the character sums S(x, χ) = Σ_{n≤x} χ(n
-
The central load-bearing assertion is the unconditional classification: 'we classify, unconditionally, all imaginary bi-quadratic and imaginary tri-quadratic fields with the Pólya index one.' If true, this completely determines those fields and is the strongest concrete result in the abstract.
-
The central claim is that there exists a sequence of short mollifiers, built from linear combinations of derivatives of $\zeta$, for which Levinson's method recovers a positive proportion of critical-line zeros uniformly as the mollifier length tends to zero. The coefficients of the linear combination are chosen as the solution of a variational problem, and the paper argues that this optimization, not the shape of the mollifier, is what keeps the proportion bounded away from zero. For modular $L$-functions, the same construction gives proportions that more than double the earlier results of Bernard and Kühn–Robles–Zeindler while using the same arithmetic moment inputs. The paper also observe
-
Assuming RH, define $P_{k/2}(T)$ as the normalized count of pairs of zeros with imaginary parts in $[T/\log^2 T, T]$ whose difference is within a small $\delta$ of $k/2$ times the average spacing $2\pi/\log T$. Under the Alternative Hypothesis for pairs (AH-Pairs), every admissible pair lies within $O((|k|+1)R(T))$ of such a half-integer, with $R(T)\to 0$. Theorem 1 shows that $1+o(1)\le P_0 \le \tfrac32 - \tfrac{2}{\pi^2} + o(1)$, and for $k\ne 0$, $P_{k/2}\sim P_0 - \tfrac12$ if $k$ is even, while $P_{k/2}\sim \tfrac32 - \tfrac{2}{\pi^2 k^2} - P_0$ if $k$ is odd. In particular, if any one limiting density $p_{k/2}$ exists, all do. Theorem 2 strengthens the error term to $R(T)\log T \to 0$
-
On the paper's own terms, the central discovery is generative rather than computational. The 1981 asymptotic formulas, which improve the 1918 Hardy–Littlewood growth exponent for $\zeta$ by roughly one-third, are presented as the engine behind a new family of $\zeta$-equivalents of the Fermat–Wiles theorem. Working through the named Jacob's ladders, the paper decomposes the Riemann zeta function into oscillatory components and then recombines them, producing statements about $\zeta$ that are claimed to be logically equivalent to Fermat's Last Theorem. If the derivation holds, those equivalences carry the weight of Fermat's Last Theorem into the analytic theory of the zeta function.
-
The paper claims that, for integers $a_1,\ldots,a_m$ with $\sum_{i=1}^m a_i=0$ and for every $h$ in a specified 'special class', the sum $$H=\sum_{0<\gamma_k\le T, 1\le k\le m} h(a_1\gamma_1+\cdots+a_m\gamma_m),$$ where the $\gamma_k$ independently run through the ordinates of the nontrivial zeros (each counted with multiplicity), satisfies a derived asymptotic formula as $T\to\infty$. The abstract states the sum and the zero-sum condition but does not display the formula's main term or the conditions defining the special class. The result is presented as a generalization of the Ford–Zaharescu theorem from one zero to $m$-tuples.
-
The central claim is that for a Galois extension $K/k$ with $k\neq \mathbb{Q}$, the discrepancy between the weighted count of unramified prime ideals of $k$ whose Frobenius symbol lies in a fixed conjugacy class $C$, and the expected proportion $|C|/|G|$ of such primes, is bounded by an expression that contains no hidden or numerically unspecified constants. Every term in the bound is an explicit function of the degree $n$, the discriminant $d_K$, the size of the conjugacy class, and the counting parameter $x$. For extensions of sufficiently small degree, the paper states a separate, sharper estimate. The proof derives this from an explicit formula for a smoothed prime-ideal counting functio
-
The abstract states: 'we prove that under certain convergence conditions on series associated to S this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provide asymptotic formulas for the corresponding prime-counting functions.' If true, this gives unconditional asymptotic formulas for the number of primes of K where the index of (G mod p) lies in S, for sets S satisfying those convergence conditions.