{"topic":"riemann-hypothesis","tier":"stated","total":198,"limit":50,"offset":0,"claims":[{"claim_id":2392539,"arxiv_id":"2608.13475","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Hidden Dyson Universality in Inverse-Spectral Geometry","paper_submitted_at":"2026-08-13T16:59:59+00:00"},{"claim_id":2699898,"arxiv_id":"2608.12257","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Power sums and Siegel-type zero-free regions for L-functions","paper_submitted_at":"2026-08-12T17:00:49+00:00"},{"claim_id":2703526,"arxiv_id":"2608.11611","paper_version":1,"claim_text":"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| $g^{{k^2+2k ell}}$.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:37:51.549630+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Upper bound for the moment of shifted values of cubic $L$-functions over function fields","paper_submitted_at":"2026-08-12T03:41:15+00:00"},{"claim_id":2703944,"arxiv_id":"2608.11520","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Contour Hankel dynamics and indicator fields for the Riemann $\\Xi$-function","paper_submitted_at":"2026-08-12T00:17:16+00:00"},{"claim_id":2355577,"arxiv_id":"2608.08714","paper_version":1,"claim_text":"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 $h^{\\mathrm{opt}}_{1,3/2}=\\sqrt{15/28}$.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples","paper_submitted_at":"2026-08-09T13:58:26+00:00"},{"claim_id":2355981,"arxiv_id":"2608.08682","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\\xi$-function","paper_submitted_at":"2026-08-09T12:58:52+00:00"},{"claim_id":1923952,"arxiv_id":"2608.07399","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Siegel zeros and small gaps between zeros of the Riemann zeta function","paper_submitted_at":"2026-08-07T16:42:11+00:00"},{"claim_id":1674147,"arxiv_id":"2608.06286","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$","paper_submitted_at":"2026-08-06T17:12:01+00:00"},{"claim_id":1921397,"arxiv_id":"2608.06435","paper_version":1,"claim_text":"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 $x$.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.75,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Chebyshev Bias for Largest Prime Factors","paper_submitted_at":"2026-08-06T06:35:58+00:00"},{"claim_id":1358825,"arxiv_id":"2608.01498","paper_version":1,"claim_text":"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)","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:15.580203+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Bounds for Mertens Sums","paper_submitted_at":"2026-08-02T21:00:11+00:00"},{"claim_id":1051882,"arxiv_id":"2607.29566","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The least quadratic residue and integers represented by quadratic forms","paper_submitted_at":"2026-07-31T15:55:33+00:00"},{"claim_id":1098581,"arxiv_id":"2607.28931","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of $-1 \\pmod N$","paper_submitted_at":"2026-07-31T01:26:20+00:00"},{"claim_id":902678,"arxiv_id":"2607.27282","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"The exceptional set of the Goldbach problem","paper_submitted_at":"2026-07-29T13:54:38+00:00"},{"claim_id":866298,"arxiv_id":"2607.26114","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Euler's $\\ell$-totients and Riemann hypothesis","paper_submitted_at":"2026-07-28T15:34:39+00:00"},{"claim_id":870781,"arxiv_id":"2607.25403","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli","paper_submitted_at":"2026-07-28T08:01:54+00:00"},{"claim_id":843447,"arxiv_id":"2607.25002","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"How Random Is the M\\\"obius Function? Smoothing, Probability, and the Riemann Hypothesis","paper_submitted_at":"2026-07-27T18:58:35+00:00"},{"claim_id":878178,"arxiv_id":"2607.22930","paper_version":1,"claim_text":"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 than T(log T)^{-A} for every fixed A. From it, Corollary 1.2 follows: for a positive density of integers x, every nontrivial finite linear combination of distinct Dirichlet L-functions is nonzero at some zero of L(s, χ0) in any sufficiently high short window.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Unshared zeros of Dirichlet $L$-functions","paper_submitted_at":"2026-07-24T22:05:47+00:00"},{"claim_id":889961,"arxiv_id":"2607.21532","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Simultaneous non-vanishing of Dirichlet $L$--functions, II: Weighted central limit theorem","paper_submitted_at":"2026-07-23T17:19:14+00:00"},{"claim_id":891745,"arxiv_id":"2607.21259","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On the Fractional Parts of Polynomials Modulo $p$","paper_submitted_at":"2026-07-23T12:31:23+00:00"},{"claim_id":895444,"arxiv_id":"2607.24830","paper_version":1,"claim_text":"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. The construction's Theorem 1.4 leaves one positive constant undetermined; the paper supplies its first numerical value, $\\mu_1=0.2347$. The Riemann zeros are not eigenvalues of this operator; the unfolded eigenvalue spacing is a rigid picket-fence, and the zeros instead appear in the explicit-formula error term of the truncated prime symbol, recovered against the true zero sum with correlation 0.9915 while shifted fake zeros fail. The operator also realizes Weil's positivity criterion: the residual of the prime symbol is bounded and $a$-independent when all zeros are on the critical line, and a single synthetic off-line zero at $\\frac12+\\delta$ is betrayed by growth of order $e^{2a\\delta}$. The authors are explicit that this proves only the easy direction of the criterion and does not approach RH.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion","paper_submitted_at":"2026-07-23T05:33:39+00:00"},{"claim_id":897351,"arxiv_id":"2607.20853","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Murmurations of quadratic Hecke $L$-functions of the Gaussian field","paper_submitted_at":"2026-07-23T02:23:53+00:00"},{"claim_id":898199,"arxiv_id":"2607.20758","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Zeros of the Dirichlet series of even zeta values","paper_submitted_at":"2026-07-22T22:19:52+00:00"},{"claim_id":899471,"arxiv_id":"2607.20408","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Extreme values of quadratic Dirichlet $L$-functions","paper_submitted_at":"2026-07-22T17:49:10+00:00"},{"claim_id":903810,"arxiv_id":"2607.20041","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On the value distribution of the Riemann zeta-function with general shifts","paper_submitted_at":"2026-07-22T11:36:24+00:00"},{"claim_id":914829,"arxiv_id":"2607.19067","paper_version":1,"claim_text":"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 $\\langle\\phi_\\varepsilon,\\eta_0\\phi_\\varepsilon\\rangle=2(1-e^{-\\varepsilon})\\to0$. Completing $C_c^\\infty(0,\\infty)$ in $\\|\\Delta\\psi\\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\\mathbb R_+)$ via $\\Phi=\\Delta\\psi$; the free self-adjoint realization is the dilation generator $D$, whose spectrum is $\\mathbb R$, simple and purely absolutely continuous. Two statements hold independently of zeta: no bounded sandwich $\\Delta^\\dagger h(D)\\Delta$ is boundedly invertible, and the transported symmetric operator $T=D|_{\\Delta D_0}$ has deficiency indices $(\\infty,\\infty)$, with the adjoint having every real point as an eigenvalue of infinite multiplicity while the free extension is purely continuous. Because $(\\Delta\\psi_z)(x)=x^{-z}$ on $x>1$, the BBM eigenfunctions on the critical line lie outside the completion, so every self-adjoint extension in this space fails to realize the proposed zeta-zero eigenfunction/boundary-condition mechanism.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Metric completion of the Bender--Brody--M\\\"uller Hamiltonian: dilation spectrum and missing eigenstates","paper_submitted_at":"2026-07-21T12:58:24+00:00"},{"claim_id":927294,"arxiv_id":"2607.17731","paper_version":1,"claim_text":"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))","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Jacob's ladders and new $\\zeta$-functionals and corresponding $\\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\\zeta$-pulses","paper_submitted_at":"2026-07-20T09:21:11+00:00"},{"claim_id":937707,"arxiv_id":"2607.16795","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"An explicit uniform cubic wedge for consecutive Toeplitz minors of the Riemann xi coefficients","paper_submitted_at":"2026-07-18T12:23:13+00:00"},{"claim_id":959241,"arxiv_id":"2607.14515","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue","paper_submitted_at":"2026-07-16T03:12:02+00:00"},{"claim_id":800820,"arxiv_id":"2607.12084","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A few remarks on the Baez-Duarte Criterion","paper_submitted_at":"2026-07-13T19:00:15+00:00"},{"claim_id":778372,"arxiv_id":"2607.09797","paper_version":1,"claim_text":"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","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Explicit formula for the discrete Laplace transform of the M\\\"obius function, related special functions, and a criterion for the Riemann hypothesis","paper_submitted_at":"2026-07-09T12:12:21+00:00"},{"claim_id":4011,"arxiv_id":"2607.05262","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Sharp Circular Sampling and Derivative Period Polynomials","paper_submitted_at":"2026-07-06T16:09:04+00:00"},{"claim_id":681011,"arxiv_id":"2607.04632","paper_version":1,"claim_text":"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}.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A decades-long breakthrough in zero-density estimates and primes in short intervals","paper_submitted_at":"2026-07-06T03:32:22+00:00"},{"claim_id":683772,"arxiv_id":"2607.04338","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A proof of Riemann's hypothesis via Hadamard-Weierstrass factorization","paper_submitted_at":"2026-07-05T14:37:00+00:00"},{"claim_id":684469,"arxiv_id":"2607.04316","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function","paper_submitted_at":"2026-07-05T14:07:00+00:00"},{"claim_id":699749,"arxiv_id":"2607.02828","paper_version":1,"claim_text":"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)\rho log T /(\\pi^{2} T).","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form","paper_submitted_at":"2026-07-02T23:39:38+00:00"},{"claim_id":11055,"arxiv_id":"2607.01703","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On Euclidean systems of ray classes","paper_submitted_at":"2026-07-02T04:52:03+00:00"},{"claim_id":17261,"arxiv_id":"2607.01126","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Distribution of Selmer ranks in prime cyclic extensions","paper_submitted_at":"2026-07-01T16:13:02+00:00"},{"claim_id":18503,"arxiv_id":"2607.00282","paper_version":1,"claim_text":"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}.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.85,"assigned_at":"2026-08-20T03:38:13.905419+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Critical Zeros and Unconditional Mean Value Theorems for twisted $\\hbox{PGL}(2)$ and $\\hbox{PGL}(3)$ $\\mathrm{L}$-functions","paper_submitted_at":"2026-07-01T00:13:25+00:00"},{"claim_id":599394,"arxiv_id":"2606.29294","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Quantum models of the Riemann zeta function, lattice spin models and algebraic models of entanglement","paper_submitted_at":"2026-06-28T09:32:18+00:00"},{"claim_id":605382,"arxiv_id":"2606.28832","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Estimating the tail of the singular product for the Hardy Littlewood and Bateman Horn conjectures","paper_submitted_at":"2026-06-27T09:37:31+00:00"},{"claim_id":574404,"arxiv_id":"2606.28227","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:37:51.549630+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Perfect powers in sequences of polygonal numbers","paper_submitted_at":"2026-06-26T16:17:48+00:00"},{"claim_id":573022,"arxiv_id":"2606.27516","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Sharp Upper Bounds for Moments of Dedekind Zeta Functions","paper_submitted_at":"2026-06-25T20:01:08+00:00"},{"claim_id":477640,"arxiv_id":"2606.25094","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Negative discrete second moments of Dirichlet $L$-functions","paper_submitted_at":"2026-06-23T18:59:58+00:00"},{"claim_id":477500,"arxiv_id":"2606.24405","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"On the Berry-Keating Operator","paper_submitted_at":"2026-06-23T10:43:48+00:00"},{"claim_id":491380,"arxiv_id":"2606.22562","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Colombeau--Beurling criterion for the Riemann hypothesis","paper_submitted_at":"2026-06-21T15:41:53+00:00"},{"claim_id":492852,"arxiv_id":"2606.24924","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Spectral Riccati--Gamma Concavity, Symmetric Zero Cancellation, and Conditional Criteria for the Riemann Hypothesis","paper_submitted_at":"2026-06-20T17:39:57+00:00"},{"claim_id":493644,"arxiv_id":"2606.22010","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:37:51.549630+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Geometry of $\\mathbf{F}_1$ and Cuntz-Krieger algebras","paper_submitted_at":"2026-06-20T12:19:21+00:00"},{"claim_id":496592,"arxiv_id":"2606.21163","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"Duke for Drinfeld","paper_submitted_at":"2026-06-19T06:57:56+00:00"},{"claim_id":518936,"arxiv_id":"2606.12376","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"phrase","confidence":0.9,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A note on a conjecture of Ng","paper_submitted_at":"2026-06-10T17:40:56+00:00"},{"claim_id":528444,"arxiv_id":"2606.10242","paper_version":1,"claim_text":"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.","claim_key":"core","tier":"stated","source":"verdict_pith","method":"signals","confidence":0.8,"assigned_at":"2026-08-20T03:38:12.374690+00:00","lean_module":null,"lean_decl":null,"lean_status":null,"reality_plus_commit":null,"paper_title":"A Tsang-range high-moment bound for $\\operatorname{Im}\\log L(\\tfrac12+it,\\chi)$ under GRH","paper_submitted_at":"2026-06-08T23:09:36+00:00"}]}