The Riemann Hypothesis

Under GRH, the authors establish 1-level density asymptotics for Hilbert modular form L-functions and their Rankin-Selberg convolutions, with applications to average orders and non-vanishing.

arxiv:2412.03034 · math.NT · 2024-12

Under GRH and the Miltersen-Vinodchandran assumption, the positivity of Schubert coefficients has a positive rule, equivalent to the problem being in NP.

arxiv:2412.18984 · math.CO · 2024-12

New error-term bounds improve Lehman's estimates for π(x) - li(x), removing the lower bound on the window width η and yielding sharper crossover regions near 10^316.

arxiv:2501.04488 · math.NT · 2025-01

For the summatory function of 2^omega(n), the paper proves an unconditional decomposition with error O(x^{8/29+epsilon}) and a conditional decomposition involving zeros of zeta(2s).

arxiv:2502.02598 · math.NT · 2025-01

For any sequence satisfying two axioms, long-term averages of its Montgomery-type form factor lie between explicit constants computed from reproducing kernels; the method refutes Gonek-Ki's conjecture for fixed c under RH.

arxiv:2502.05106 · math.NT · 2025-02

For prime moduli p, the smoothed m-th moment of quadratic twists of a fixed modular form's coefficients grows like X Y^{m/2} (log X)^{m(m-3)/2}, with matching even-m lower bounds, conditional on GRH.

arxiv:2608.05961 · math.NT · 2026-08

Explicit Wasserstein-distance rates are proved for equidistribution of ultra-short exponential sums and of Deligne-Katz trace-function families over finite fields.

arxiv:2505.22059 · math.NT · 2025-05

A regulated area-integral convergence criterion is claimed to prove RH, but the proof's key divergence estimate is contradicted by the paper's own excision of zero neighborhoods.

arxiv:2505.23238 · math.GM · 2025-05

An explicit formula expresses the exceptional-set exponent for the short-interval prime number theorem in terms of zero density estimates, yielding improved numerical bounds such as μ(17/30) ≤ 7/12.

arxiv:2505.24017 · math.NT · 2025-05

For every k>2, unconditionally, the average of |∑_{n≤x} n^{-it}|^{2k} over t∈[0,T] is ≫_k x^k (log L)^{(k-1)^2}, where L = min{x, T/x}.

arxiv:2506.09334 · math.NT · 2025-06

The joint moments of log ζ and a smoothed zero-counting function factorize into Gaussian moments, showing asymptotic independence.

arxiv:2507.04150 · math.NT · 2025-07

Under GRH, in every Galois orbit modulo p^k, a positive proportion of characters chi satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0 for distinct imprimitive twists eta1, eta2.

arxiv:2507.06609 · math.NT · 2025-07

If the Alternative Hypothesis plus the new weak density condition holds, then asymptotically 100% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line.

arxiv:2507.06823 · math.NT · 2025-07

Riemann sequences are defined to mimic the zeta-zero asymptotic; the zeta zeros form one, infinitely many others are constructed via crystalline measures, and the paper asks whether the zeta sequence is the unique real one.

arxiv:2507.07253 · math.NT · 2025-07

For the equation x^2 - 2 = y^p, the authors prove the only solutions for p>911 are y = -1, and any nontrivial solution has y > 10^1000.

arxiv:2507.12397 · math.NT · 2025-07

At least 33% of the special values L(1/2+it_f, f) of Hecke-Maass L-functions are nonzero, a new effective non-vanishing record.

arxiv:2507.14566 · math.NT · 2025-07

A generating-function framework solves convolution-like recurrences and constructs b-sequences whose alpha-sequences are partial sums of zeta values, pi, e, and many OEIS sequences.

arxiv:2507.23619 · math.NT · 2025-07

Extends prime ideal race bias formulas to r competing conjugacy classes and proves that for r ≥ 3 the moderacy criterion no longer depends on r.

arxiv:2508.04087 · math.NT · 2025-08

New explicit exponential- and log-type bounds for Mertens sums over primes, powered by bounds for weighted zero sums of the Riemann zeta function, improving on Vanlalngaia (2017) and correcting Dusart (2018).

arxiv:2608.01498 · math.NT · 2026-08

An asymptotic formula for Σ h(a_1γ_1+...+a_mγ_m) over m-tuples of nontrivial zeta zeros with integer coefficients summing to zero, generalizing the Ford-Zaharescu theorem.

arxiv:2508.18280 · math.NT · 2025-08

The paper produces asymptotic counting formulas for w-coprime S-integers and S-integral ideals in function fields over finite fields, proved with Riemann-Roch and the Weil theorem.

arxiv:2508.10484 · math.NT · 2025-08

Under GRH, the authors bound shifted moments of cubic and quartic Dirichlet L-functions and, as a corollary, moments of the corresponding character sums.

arxiv:2508.14534 · math.NT · 2025-08

Under GRH, global Bessel periods for symmetric cubes of GL(2) forms are asymptotically decorrelated when averaged over imaginary quadratic fields.

arxiv:2508.15964 · math.NT · 2025-08

The claimed Salem-type equivalence for the Riemann hypothesis is not established; the proof assumes the hypothesis in the crucial step.

arxiv:2508.21789 · math.NT · 2025-08

Under GRH, a prime-level newform's mass pushed through the Hecke correspondence equidistributes on Y1 × Y1 at rate (log q)^{-1/4+ε}, confirming the authors' new joint conjecture.

arxiv:2509.01602 · math.NT · 2025-09

New conjectures give the full asymptotic expansion of discrete moments of mixed derivatives of the Riemann zeta function at its zeros, including a previously unknown formula for the second moment of ζ'(ρ).

arxiv:2509.07792 · math.NT · 2025-09

For Re(k)>-3, the average of zeta'(1/2+i gamma)^k over zeros is conjectured to be (1/Gamma(k+2)) (log(T/2pi))^k.

arxiv:2509.07788 · math.NT · 2025-09

For Rademacher random multiplicative functions, the paper attempts to show that S_n/a_n has no weak limit for sub-square-root normalizations, conditional on an unverified counting assumption, and claims new moment and concentration bounds; but key proofs contain invalid steps.

arxiv:2509.19067 · math.PR · 2025-09

For large enough finite fields, every residue class modulo a squarefree degree-n polynomial is claimed to be a product of two monic irreducible polynomials of degree at most n, via a one-dimensional Katz equidistribution proof.

arxiv:2510.17612 · math.NT · 2025-10

For weighted projective stacks over F_q, the number of F_q-orbits on nonzero representatives is sum over nonempty coordinate supports S of (q-1)^{|S|-1} times gcd(k_S, q-1), where k_S is the gcd of the weights in S.

arxiv:2511.12812 · math.AG · 2025-11

The paper reduces the Riemann hypothesis to an unproved sign inequality for a differential equation driven by the fractional part, then asserts the inequality without proof.

arxiv:2511.13496 · math.DS · 2025-11

Under a strong unproved 'Deep Riemann Hypothesis', the paper claims a deterministic prime-bias ranking where the residue class -1 mod N dominates, governed by log L(1, χ).

arxiv:2607.28931 · math.NT · 2026-07

The paper's claimed Riemann-hypothesis bound for the analytic part of the divisor summatory error term contradicts its own definition of that analytic part.

arxiv:2601.11052 · math.NT · 2026-01

Under RH, the paper derives new explicit bounds for Re(ζ'/ζ), |ζ|, |1/ζ|, and |ζ'/ζ| on Re(s)=1, improving known lower-order constants.

arxiv:2602.06199 · math.NT · 2026-02

Conditional on GRH and negative moment bounds, e^{-y/(2k)}∑_{n≤e^y} μ^(k)(n)χ(n) (or a modified character) has a logarithmic limiting distribution, and the sums obey an almost-all O(x^{1/(2k)}(log x)^{1/2+ε}) bound.

arxiv:2602.23100 · math.NT · 2026-02

If the Laplace-weighted Möbius sum Σ μ(n)e^{-nx} is O(x^{-1/2}) as x→0+, the Riemann hypothesis holds; an explicit formula with new double-pole terms is proved under a simplicity assumption.

arxiv:2607.09797 · math.GM · 2026-07

Conditional on GRH and modularity, the m-th moment of analytic rank of elliptic curves over a degree-d number field is bounded by O((9dm/2)^m), with a corresponding exponential tail bound.

arxiv:2607.15998 · math.NT · 2026-07

The paper restates Moser's ζ-functional as sums over zero-to-zero intervals and derives 'ζ-equivalents' of Fermat's Last Theorem that are tautological consequences of the claimed asymptotic.

arxiv:2607.17731 · math.NT · 2026-07

The Riemann zeta-function is shown to have the universality and Bohr–Jessen distribution properties under logarithmic-power shifts (log t)^α, α>1, with quantitative discrepancy bounds.

arxiv:2607.20041 · math.NT · 2026-07

RH is equivalent to subpolynomial growth of arbitrarily high finite local moments of normalized Möbius polynomials on arcs of radius c/N.

arxiv:2607.25002 · math.PR · 2026-07

Local reduction probabilities for genus-g hyperelliptic curves with a Weierstrass point are determined, yielding a conditional explicit upper bound on their average analytic rank.

arxiv:2607.12381 · math.NT · 2026-07

Freely decaying incompressible turbulence possesses a universal Euler-ensemble attractor whose continuum Mellin spectrum is controlled by the non-trivial zeros of the Riemann zeta function, producing an infinite-time essential singularity under RH.

arxiv:2604.12207 · hep-th · 2026-04

RH is equivalent to a single moderate net from damped Báez–Duarte sums (via Mellin convolution) being uniformly L²-bounded and associated with −χ_{(0,1)} in G(0,1).

arxiv:2606.22562 · math.CV · 2026-06

Under RH and a pole-damping width bound, a Gaussian–Perron prime-side defect near each fixed simple critical-line zero equals −a Re(e^{−λ}/λ) up to an exponentially small nonlocal error.

arxiv:2607.04316 · math.NT · 2026-07

Formulates Euclidean systems for ray classes, proves they generate the ray class group, and shows under GRH that generating sets of Cl_K^{(p)^N} are Euclidean systems for specified totally real Galois fields.

arxiv:2607.01703 · math.NT · 2026-07

Modifications to Klagsbrun-Mazur-Rubin yield bounds on Selmer rank distributions for Galois modules in p-cyclic extensions ordered by ramified primes, with applications to elliptic curve rank gains and point counts on superelliptic and hyperelliptic curves.

arxiv:2607.01126 · math.NT · 2026-07

The paper proves L(n) = O(n log log n / log n) and B(n) ≥ c log n unconditionally, plus under RH the bounds B(n) = O(n^{3/4} (log n)^{1/2}) and L(n) ≥ c' n^{1/4} (log n)^{-1/2}.

arxiv:2605.22752 · math.NT · 2026-05

If the θ=∞ conjecture holds for automorphic L-functions, then they are non-vanishing in certain critical strip regions and families satisfy a quasi-Riemann Hypothesis.

arxiv:2605.24363 · math.NT · 2026-05

Rules out naive concavity criterion for RH and derives conditional zero-density and localisation criteria via a finite spectral Riccati-Gamma averaging framework.

arxiv:2606.24924 · math.GM · 2026-06

A universal estimate shows the singular product tail for Hardy-Littlewood and Bateman-Horn conjectures decays as 1/log regardless of system structure, with superfast convergence for linear cases and Galois-averaged coefficients for nonlinear ones under RH.

arxiv:2606.28832 · math.GM · 2026-06

Overview of quantum models linking the Riemann zeta function to the Hilbert-Polya conjecture, lattice spin models for entanglement, and p-adic quantum computing.

arxiv:2606.29294 · math.NT · 2026-06

Constructs Dixmier traces via eigenvalue sequences in weak Lorentz ideals, gives spectral characterization of measurable operators answering Connes, and applies to nonclassical Weyl laws.

arxiv:2605.25591 · math.OA · 2026-05

Sign changes of multiplicative functions occur in every reduced residue class mod q by size q^{2+o(1)} unless the function strongly pretends to be a real Dirichlet character mod q.

arxiv:2605.27833 · math.NT · 2026-05

Extends complete solution lists for perfect-power polygonal numbers to s=2k+4 and s=k+4 (k prime in specified ranges) via modular, hypergeometric, and linear-forms methods plus computations.

arxiv:2606.28227 · math.NT · 2026-06

Under GRH, the order of magnitude of E|sum_{n≤x} h(n) λ(n)|^{2q} is determined up to e^{O(q^2)} for 1 ≤ q ≤ c log x / log log x.

arxiv:2606.01514 · math.NT · 2026-06

A framework via the screw function unifies work on the Weil quadratic form and yields a conjecture on a spectral operator for the imaginary parts of nontrivial zeta zeros, without assuming the Riemann hypothesis.

arxiv:2606.09096 · math.NT · 2026-06

Conditional lower bound (half the conjectured value) for the second moment of a zeta ratio summed over non-trivial zeros, assuming RH and simple zeros.

arxiv:2606.12376 · math.NT · 2026-06

Proves equidistribution of CM points on Drinfeld-Stuhler modular curves by reducing via Weyl criterion to toric period decay, bounded using Waldspurger's formula and the Riemann hypothesis over function fields.

arxiv:2606.21163 · math.NT · 2026-06

The paper maps projective varieties over F1 to Cuntz-Krieger algebras O_A, uses their K-theory to compute Frobenius actions and cardinalities, verifies most Weil conjectures for the zeta function, and constructs a morphism Spec(Z) to Spec(F1) via crossed products.

arxiv:2606.22010 · math.NT · 2026-06

Assuming GRH and simple zeros, lower bounds are established for ∑ |L'(ρ,χ)|^{-2} and ∑ |L(2ρ,χ²)/L'(ρ,χ)|² that recover half the conjectured asymptotic when q is fixed and degrade to 1/(2+A) when q = T^A.

arxiv:2606.25094 · math.NT · 2026-06

A review presenting two complementary mathematical viewpoints on the Berry-Keating operator in the context of its possible link to the Riemann hypothesis.

arxiv:2606.24405 · math-ph · 2026-06

Derives the complete asymptotic polynomial for the mean value of the μ-th derivative of quadratic Dirichlet L-functions over the rational function field.

arxiv:1906.10373 · math.NT · 2019-06

Sums of zeta^{(n)}(rho) over non-trivial zeros rho admit a full asymptotic expansion whose leading term is real and positive for odd n, negative for even n.

arxiv:2106.03005 · math.NT · 2021-06

The inequality log g(n) < li^{-1}(n) for all n > 0 is equivalent to the Riemann hypothesis.

arxiv:1907.07664 · math.NT · 2019-07

Claims to disprove that the closure of real parts of non-trivial zeta zeros has measure at least 0.5 and to prove the Lindelöf hypothesis via pseudodifferential arithmetic applied to an operator built from zeta zeros.

arxiv:2208.12937 · math.NT · 2022-08

Claims a simple proof that the non-trivial zeros of the Riemann zeta function all have real part equal to 1/2.

arxiv:2209.01890 · math.GM · 2022-09

Under GRH the authors prove existence of a positive natural density for primes whose reduction of an abelian variety satisfies a bounded cyclic-component condition on the quotient by given points together with a Frobenius condition.

arxiv:2212.03386 · math.NT · 2022-12

Using an associated Galois representation, the authors construct examples of globally primitive points on elliptic curves that are locally imprimitive at primes of cyclic reduction.

arxiv:2304.03964 · math.NT · 2023-04

Conditional on GRH for quadratic L-functions, proves explicit uniform spectral gap for Laplacian on X_0(p) covers of Schottky surfaces X when limit set is thick enough.

arxiv:2305.02228 · math.SP · 2023-05

Claims an unconditional proof that π(x + x^λ) − π(x) ∼ x^λ / log(x) holds for all 0 < λ < 1.

arxiv:2307.08725 · math.NT · 2023-07

Under RH and simple zeros, the correlations equal -3/π²(1-T^{(c-1)δ(T)}) + sum 1/|ρ ζ'(ρ)|² for μ/M and a similar explicit sum for λ/L.

arxiv:2409.02106 · math.NT · 2024-09

Further results on properties of the expected value of a general size-biased distribution related to the Riemann xi-function, linked to functional equations and the Riemann hypothesis.

arxiv:2412.19347 · math.NT · 2024-12

The Hartle-Hawking state for toroidal quantum cosmologies is expressed in the Langlands decomposition as a sum over zeta zeros whose near-singularity dynamics follow the Hilbert-Pólya Hamiltonian and as a Möbius average of CFT partition functions.

arxiv:2505.03068 · hep-th · 2025-05

A discrete spectral framework on cyclic graphs produces exact identities for special values of Dirichlet L-functions at integers and reformulates GRH for odd primitive characters via asymptotic functional equations.

arxiv:2512.01779 · math.NT · 2025-12

Extends one-level density support to (-8/3, 8/3) for the unitary family of Gamma_1(q) L-functions under GRH, verifying the Katz-Sarnak prediction and yielding a 62.5% central non-vanishing proportion.

arxiv:2512.20066 · math.NT · 2025-12

Proves asymptotic for the number of ternary Goldbach representations of large odd N with one prime bounded by U = N^{4/49} exp(log^{2/3+ε}N) unconditionally or log^{4+ε}N under GRH.

arxiv:2605.19566 · math.NT · 2026-05

Under GRH, one-level densities for even and odd orthogonal families of automorphic L-functions are established with Fourier support extended to (-3,3), giving the strongest conditional non-vanishing results at the central point.

arxiv:2605.17012 · math.NT · 2026-05

The paper proves equivalences between the Riemann hypothesis and persistent inequalities of normalized error terms in weighted prime counting functions, and computes conditional logarithmic densities of sign agreements such as ≈0.9865 for specific pairs.

arxiv:2507.13504 · math.NT · 2025-07

A Fibonacci-Redheffer matrix is defined and analyzed, yielding determinant and spectral results plus a new expression related to the Riemann hypothesis.

arxiv:2511.13627 · math.NT · 2025-11

Unconditional construction of moduli sequences forces generalized Skewes numbers to grow rapidly in quadratic-residue races, disproving Fiorilli's conjecture, with conditional upper bounds under GRH and effective linear independence via quantitative Wasserstein convergence rates.

arxiv:2603.20093 · math.NT · 2026-03

For quadratic twists of elliptic curves over Q, the Taylor coefficients of their L-functions at the central point are non-vanishing for large discriminants under GRH, with an unconditional lower bound on the number of non-vanishing ones derived from moments of central values.

arxiv:2605.09251 · math.NT · 2026-05

The resonance-correlation method proves μ < 0.50895 for the liminf of normalized gaps between consecutive Riemann zeta zeros under the Riemann Hypothesis.

arxiv:2604.05733 · math.NT · 2026-04

For almost all primes p, the largest prime factor of a_f(p) + a_g(p) for twist-inequivalent non-CM newforms f,g exceeds (log p)^{1/14} (log log p)^{3/7-ε}, with a similar density-one result for integers and an exponential lower bound under GRH.

arxiv:2604.07474 · math.NT · 2026-04

Under GRH, four twisted Dirichlet L-functions at 1/2 + it are simultaneously non-zero for a positive proportion of characters χ mod q (q prime, sufficiently large); unconditionally, this holds for infinitely many χ but with proportion tending to zero.

arxiv:2604.11941 · math.NT · 2026-04

The authors present an equivalence of the Riemann Hypothesis to a property of the Salem integral equation.

arxiv:2604.15396 · math.GM · 2026-04

For quadratic fields with 3-class group (Z/3Z)^2 and given 3-principalization types, the paper states exact criteria for metabelian 3-class towers of length 2 or 3 and computes minimal discriminants experimentally under GRH for nilpotency classes up to 11.

arxiv:2605.01590 · math.NT · 2026-05

Every sufficiently large integer can be written as the sum of a prime and a non-square-free integer.

arxiv:2605.02426 · math.NT · 2026-05

Probabilistic independence assumptions on Ω(n) and μ²(n) recover the limiting asymptotic growth of |M(x)| / √x with hypotheses claimed to be more attainable than the Riemann Hypothesis or linear independence of zeta zeros.

arxiv:2604.23517 · math.NT · 2026-04

Under RH, the measure of t in [T,2T] with |zeta(1/2+it)| > (log T)^k is <= C_k (log T)^{-k^2}/sqrt(log log T) with C_k=exp(e^{ck}), implying 2k-moment bounds C_k (log T)^{k^2}.

arxiv:2604.25579 · math.NT · 2026-04

Any number of primitive GL(1) and GL(2) L-functions can simultaneously take large values on the critical line unconditionally, improving prior conditional results.

arxiv:2605.03665 · math.NT · 2026-05