The Quasi-Riemann Hypothesis

openai/math repo 的第 003 項結果，證明所有 Dirichlet $L$-function（包含 Riemann zeta function $\zeta(s)$）在半平面 $\mathrm{Re}(s)>7/8$ 沒有零點。同一個半平面對 $\mathbb{Q}(\sqrt{-3})$ 上的每個 finite-order Hecke $L$-function 也成立。

相關的 OpenAI preprint 有三份：

主文：The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$ preprint（2026-09-30）
Landau–Siegel zeros：Uniform exclusion of Landau–Siegel zeros preprint（2026-10-01）
$\mathrm{Re}(s)>11/12$ 的證明：The Quasi-Riemann Hypothesis preprint（2026-10-05）

Quasi-Riemann hypothesis

對 $\zeta(s)$ 而言，存在固定的 $\theta_0<1$，使 $\zeta(s)$ 在 $\mathrm{Re}(s)>\theta_0$ 沒有零點，稱為 quasi-Riemann hypothesis。它要求的是在每個高度都成立的同一個間隙，而不只是 $\mathrm{Re}(s)=1$ 上不為零。對 Dirichlet $L$-function 的對應版本，要求單一的 $\theta_0<1$ 對所有 character 一致，與 conductor 及高度無關。

古典的 zero-free region 隨 conductor 或高度增大而逼近 $\mathrm{Re}(s)=1$，且可能允許一個單一的 real exceptional zero，因此無法給出固定半平面的一致排除。Zero-density estimate 則是估計位於某條垂直線右側的零點個數，並不排除所有零點。

OpenAI #003

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$ preprint

Abstract

We prove that all finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ and all Dirichlet $L$-functions are zero-free in the half-plane $\mathrm{Re}(s)>7/8$, with the principal pole at $s=1$ allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.

Main theorem

Theorem 1.1 Every finite-order Hecke $L$-function over $F=\mathbb{Q}(\sqrt{-3})$ has no zero in $\mathrm{Re}(s)>7/8$. The same holds for every Dirichlet $L$-function, including $\zeta(s)$. A pole at $s=1$ for a principal character is allowed.

Companion preprints

11/12 的證明

The Quasi-Riemann Hypothesis preprint

We establish the quasi-Riemann hypothesis by proving that every Dirichlet $L$-function, including Riemann’s zeta function, has no zeros in the half-plane $\mathrm{Re}(s)>11/12$. More generally, we prove the same zero-free half-plane for every finite-order Hecke $L$-function over $K=\mathbb{Q}(\sqrt{-3})$. In particular, this rules out the existence of Landau–Siegel zeros.

Theorem 1.1 Every finite-order Hecke $L$-function over $K$ has no zeros in the half-plane $\mathrm{Re}(s)>11/12$. Thus every Dirichlet $L$-function, including $\zeta(s)$, has no zeros for $\mathrm{Re}(s)>11/12$.

Remark. “Theorem 1.1 is a natural intermediate step toward the stronger zero-free region Re s > 7/8 established in [36, Theorem 1.1]. For simplicity, we have isolated Theorem 1.1 and its proof here.” “Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability.”

Landau–Siegel zeros

We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant $c>0$ such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$-function of conductor $q\ge3$ satisfies $(1-\beta)\log q\ge c$.

Theorem 1 There is an absolute constant $c>0$ such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$-function of conductor $q\ge3$ satisfies $$(1-\beta)\log q\ge c.$$

驗證狀態

openai/math 的結果處於不同的驗證階段，沒有 formalization 的結果可能有問題。#003 在 repo 中有對應的 Lean formalization docs，對應的論文是 7/8 主文與 Landau–Siegel zeros 的 preprint。

Comparator statement

Result	Comparator statement
Riemann zeta $7/8$ bound	QuasiRiemannHypothesis.lean
Dirichlet $L$-function $7/8$ bound	DirichletSevenEighths.lean
Finite-order Hecke $L$-function $7/8$ bound	HeckeSevenEighths.lean
Uniform real-zero gap	SiegelZeros.lean