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Math-2797 · mathematics · Number theory

The eleven-twelfths zero-free half-plane

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Do all Dirichlet L-functions, including zeta, and all finite-order Hecke L-functions over Q(sqrt(-3)) have no zeros in $Re(s)>11/12$, with any principal-character pole at one allowed?

Short English statement

Do all Dirichlet L-functions, including zeta, and all finite-order Hecke L-functions over Q(sqrt(-3)) have no zeros in $Re(s)>11/12$, with any principal-character pole at one allowed?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2797-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2797

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 003; 2026-10-05).

The eleven-twelfths zero-free half-plane

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 $11/12$.

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.

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.

Formal verification

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

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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