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Math-2474 · mathematics · Number theory

Spectral-Deformation Methods for the Riemann Hypothesis

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Can the spectral-deformation method based on $H_\lambda(t)$ prove the Riemann hypothesis despite the failure of the analogous assertion for $H_0(t)$ in general?

Formulation references: [S1], [S2].

Short English statement

Can the spectral-deformation method based on $H_\lambda(t)$ prove the Riemann hypothesis despite the failure of the analogous assertion for $H_0(t)$ in general?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2474-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2474

The linked results assert zero-free half-planes and a uniform gap for real Dirichlet zeros near one. They neither analyze the spectral deformation $H_\lambda(t)$ nor establish that this method proves the critical-line statement, so their connection to this method-level question is indirect.

Spectral-Deformation Methods for the Riemann Hypothesis

OpenAI's claim

The linked manuscripts claim zero-free regions for their specified zeta, Dirichlet and Hecke $L$-functions, together with a uniform exclusion of exceptional real Dirichlet zeros near one. These results concern the same $L$-functions as the Riemann hypothesis and are related to the spectral-deformation question. None asserts a bound on the de Bruijn--Newman constant for $H_\lambda(t)$ or claims that this deformation proves the Riemann hypothesis.

Mathematical statement of the claimed result

Manuscript 1.
For $K=\mathbb Q(\sqrt{-3})$ and every finite-order Hecke character $\chi$ over $K$, the claimed zero-free assertion, with meromorphic poles excluded from evaluation, is \[ \operatorname{Re}s>11/12\quad\Longrightarrow\quad L_K(s,\chi)\ne0. \] It includes every Dirichlet $L$-function and the Riemann zeta function. The accompanying prime-progression claim uses $\pi(x;q,a)$ and asserts, with an absolute effective constant, \[ \sup_{\substack{1\le q\le x\$a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)}\int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x\quad(x\ge2). \] The critical-line requirement $\operatorname{Re}\rho=1/2$ is not a consequence of this half-plane claim.
Manuscript 2.
For every finite-order Hecke character of $F=\mathbb Q(\sqrt{-3})$, every Dirichlet character and the zeta function, the principal claim is \[ \operatorname{Re}s>7/8\quad\Longrightarrow\quad L(s)\ne0, \] where a principal character's pole at $s=1$ is allowed. The weaker displayed $11/12$ theorem follows within this scope. The cited arithmetic consequence gives absolute $A,C>0$ such that every odd prime $p$ has least positive quadratic nonresidue \[ n(p)\le C(\log p)^A. \] Thus $n(p)\ll_{\delta} p^{\delta}$ for each fixed $\delta>0$. Given binary $p$ and $a\in\mathbb F_p$, a deterministic algorithm of time polynomial in $\log p$ is further claimed to return a square root of $a$ or correctly declare it nonexistent. The half-plane assertion leaves zeros between the critical line and $7/8$ untreated.
Manuscript 3.
The manuscript's Theorem 1 asserts an absolute positive constant, uniform over all real primitive characters: \[ \exists c>0\ \forall q\ge3\ \forall\chi\text{ primitive, real, nonprincipal of conductor }q \ \forall\beta\in(0,1):\quad L(\beta,\chi)=0 \ \Longrightarrow\ (1-\beta)\log q\ge c. \] Thus these real zeros cannot approach one faster than a constant multiple of $1/\log q$. The assertion concerns real Dirichlet zeros; it does not place all nonreal zeros on the critical line.

Outline of the argument

For manuscript 1 (family 003; 2026-10-05), twisted Möbius sums are enlarged to a family before Poisson summation. A recursive estimate for the dual mean square, followed by inversion of cube factors, supplies the saving used to exclude a reciprocal-function singularity.

For manuscript 2 (family 003; 2026-09-30), smoothed character sums are estimated in two ways: directly by large-sieve and recursive moment bounds, and through a Mellin transform involving the reciprocal of an $L$-function. Poisson summation and sextic residue characters align the two expressions so that their savings force continuation of the reciprocal.

For manuscript 3 (family 003; 2026-10-01), a hypothetical exceptional zero forces a pronounced prime bias. Interpolation on a projected integer box produces a weighted conjugate determinant whose Archimedean size conflicts with its divisibility at primes selected by that bias.

Formal verification

For manuscript 2 (family 003; 2026-09-30), OpenAI lists Lean statements for this assertion: zeta, Dirichlet $L$-functions, and finite-order Hecke $L$-functions for $\mathbb Q(\sqrt{-3})$ have no zeros with real part greater than $7/8$, with poles excluded.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 3 (family 003; 2026-10-01), OpenAI lists Lean statements for this assertion: an absolute positive constant bounds $(1-\beta)\log q$ away from zero for real zeros of primitive nonprincipal real Dirichlet characters of conductor $q\ge3$.

This coverage does not establish a formal proof of the complete problem as stated here.

No Lean formalization is listed at this snapshot for manuscript 1 (family 003; 2026-10-05).

The linked family Comparator checks are DirichletSevenEighths.lean, HeckeSevenEighths.lean, QuasiRiemannHypothesis.lean, SiegelZeros.lean.

Their named declarations include OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SiegelZeros.WeightedTorusJets.dirichletRealZeroBound_proof, OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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