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Math-2253 · mathematics · Number theory

The density hypothesis for zeroes of the Riemann zeta function

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
Let $N(\sigma,T)$ count zeros $\rho=\beta+i\gamma$ of the Riemann zeta function with $\beta\geq\sigma$ and $0<\gamma\leq T$, counting multiplicity. For every $1/2\leq\sigma\leq1$ and every $\varepsilon>0$, must \[N(\sigma,T)\ll_{\sigma,\varepsilon}T^{2(1-\sigma)+\varepsilon}\] hold as $T\to\infty$?

Formulation references: [S1], [S2].

Short English statement

Let $N(\sigma,T)$ count zeros $\rho=\beta+i\gamma$ of the Riemann zeta function with $\beta\geq\sigma$ and $0<\gamma\leq T$, counting multiplicity. For every $1/2\leq\sigma\leq1$ and every $\varepsilon>0$, must \[N(\sigma,T)\ll_{\sigma,\varepsilon}T^{2(1-\sigma)+\varepsilon}\] hold as $T\to\infty$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2253-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2253

The requested density exponent must hold across the full range starting at real part $1/2$. A zero-free half-plane makes the count vanish only above its fixed boundary, so it supplies no density estimate in the rest of that range.

The density hypothesis for zeroes of the Riemann zeta function

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 003; 2026-10-05), the asserted result is that zeta, Dirichlet functions, and finite-order Eisenstein-field Hecke families admit a zero-free region with boundary $11/12$; In manuscript 2 (family 003; 2026-09-30), the asserted result is that 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.

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.

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.

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.

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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