Motivation

The Riemann zeta function $\zeta(s) = \sum_{n\ge1} n^{-s}$ ($\Re s > 1$), continued meromorphically to $\mathbb{C}$, encodes the distribution of primes through its zeros. Riemann (1859) conjectured that all nontrivial zeros lie on $\Re s = 1/2$. Hadamard and de la Vallée Poussin (1896) proved there are no zeros on $\Re s = 1$, which gives the prime number theorem, but all classical zero-free regions shrink toward the line $\Re s = 1$ as the height grows. The quasi-Riemann hypothesis asks for a fixed $\sigma_0 < 1$ such that $\zeta(s) \ne 0$ for all $\Re s > \sigma_0$: one zero-free gap valid at every height. The uniform version for all Dirichlet $L$-functions is a standard formulation in analytic number theory (e.g. Friedlander–Goldston).

This preprint claims a proof with $\sigma_0 = 7/8$, for $\zeta$, for all Dirichlet $L$-functions, and for all finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$.

Background

• 1837. Dirichlet introduces character $L$-series to prove infinitude of primes in arithmetic progressions.
• 1859. Riemann relates zeros of $\zeta$ to primes and formulates the Riemann hypothesis.
• 1896. Hadamard and de la Vallée Poussin prove $\zeta(s) \ne 0$ on $\Re s = 1$.
• 1918. Hecke develops $L$-functions of characters over number fields.
• 1969, 1977. Kubota's metaplectic theory and Patterson's cubic theta series supply the automorphic objects used here.
• 2000–2014. Heath-Brown's cubic large sieve; Goldmakher–Louvel's quadratic large sieve over number fields; Blomer–Goldmakher–Louvel's higher-order recursion.
• 2026. Guth and Maynard improve zero-density bounds for $\zeta$ (which bound the number of zeros but do not exclude them). The OpenAI preprint The Quasi-Riemann Hypothesis (dated September 30, 2026) claims the zero-free half-plane $\Re s > 7/8$. It has not been peer reviewed or independently verified; the Lean goal is open on this platform. The Riemann hypothesis itself is not claimed.

Setting

For a Dirichlet character $\chi$ modulo $q$ (extended by zero on non-units), $L(s,\chi) = \sum_{n \ge 1}\chi(n) n^{-s}$ for $\Re s > 1$, continued meromorphically; the character modulo $1$ gives $\zeta$. For $F = \mathbb{Q}(\sqrt{-3})$ with ring of integers $\mathcal{O}$, a finite-order Hecke character modulo a nonzero ideal $\mathfrak{f}$ is a character of the ray class group modulo $\mathfrak f$, extended by zero, and $L_F(s,\eta) = \sum_{\mathfrak a \ne 0}\eta(\mathfrak a)(N\mathfrak a)^{-s}$. Principal characters have a pole at $s = 1$.

In Lean, $\zeta$ is Mathlib's riemannZeta, Dirichlet $L$-functions are Mathlib's DirichletCharacter.LFunction, and the Hecke family is OAI.SevenEighths.HeckeFamily.LFunction: a Character is a multiplicative character of $\mathcal O/\mathfrak m$ trivial on units (for this class-number-one field these are exactly the ray class characters), and its $L$-function is a lattice sum over $\mathcal{O} = \mathbb{Z}[\omega]$ divided by the $6$ units, continued through a theta-type functional equation pair.

Formalization targets

Goal: Theorem 1.1 for $\zeta$ (quasi-Riemann hypothesis)

$$\Re s > 7/8 \implies \zeta(s) \ne 0 .$$

Milestone: Theorem 1.1 for Dirichlet $L$-functions

For every Dirichlet character $\chi$ modulo $q \ge 1$ and every $s$ with $\Re s > 7/8$, $L(s,\chi) \ne 0$, except at the pole $s = 1$ of a principal character.

Milestone: Theorem 1.1 for Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$

For every finite-order Hecke character $\eta$ of $F = \mathbb{Q}(\sqrt{-3})$ and every $s$ with $\Re s > 7/8$, $L_F(s,\eta) \ne 0$, except at the pole $s = 1$ of a principal character.

The three statements are ordered from weakest to strongest; the proof in the preprint goes through the whole Hecke family even for $\zeta$, because Poisson summation introduces Hecke twists of the target.

Significance

The result. A fixed zero-free half-plane would give power-saving error terms in the prime number theorem and in prime counts in arithmetic progressions, uniformly in the modulus, and would exclude Siegel zeros beyond $7/8$. The preprint derives (Corollary 1.2) that the least quadratic nonresidue modulo $p$ is $O((\log p)^A)$, proving Vinogradov's conjecture, and a deterministic polynomial-time square-root algorithm modulo primes.

Formalizing it. This is a claimed resolution of a long-standing question in an unrefereed preprint; a machine-checked proof would be strong independent verification. The goal uses only Mathlib definitions, so it cannot be weakened by a definitional choice. The Hecke milestone uses a custom $L$-function definition that must also be checked against the source.

Difficulty

Classical zero-free regions come from positivity arguments (the $3 + 4\cos\theta + \cos 2\theta$ inequality) that give regions of width $c/\log(|t|+q)$ shrinking to the line $\Re s = 1$; they cannot produce a fixed half-plane. Zero-density estimates bound the number of zeros to the right of a line but do not exclude every zero. The preprint instead continues reciprocals $1/L$ across a uniform distance by comparing two exact representations of a completed cubic-theta sum, with exponent margins independent of the target character.

Formalization scope

• The goal: ∀ s : ℂ, 7/8 < s.re → riemannZeta s ≠ 0. Mathlib assigns $\zeta(1)$ a finite nonzero value, so no pole exclusion is needed.
• The Dirichlet milestone excludes exactly the case χ = 1 ∧ s = 1, where Mathlib's value is a junk value at the pole.
• The Hecke milestone excludes χ.residue = 1 ∧ s = 1.
• The boundary $\Re s = 7/8$ is not included.

The goal is not vacuous and cannot be satisfied by a convention: it is a statement about Mathlib's riemannZeta on an open half-plane.

Needed infrastructure: Hecke $L$-functions over imaginary quadratic fields and their functional equations, cubic and sextic residue symbols, Patterson's cubic theta coefficients, large sieve inequalities over $\mathbb{Z}[\omega]$, and Mellin inversion. Contributions toward the $11/12$ half-plane (Theorem 3.1) as an intermediate target are welcome.

Selected references

• OpenAI, The Quasi-Riemann Hypothesis, OpenAI Math Release preprint, September 30, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf
• B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Königl. Preuss. Akad. Wiss. Berlin (1859).
• J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896).
• E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 1 (1918).
• J. B. Friedlander, D. A. Goldston, Sums of three or more primes, Trans. Amer. Math. Soc. 349 (1997).
• K. Billington, M. Cheng, J. Schettler, A. I. Suriajaya, The average number of Goldbach representations and zero-free regions of the Riemann zeta-function, Int. J. Number Theory 21 (2025).
• S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977).
• D. R. Heath-Brown, Kummer's conjecture for cubic Gauss sums, Israel J. Math. 120 (2000). https://doi.org/10.1007/s11856-000-1273-y
• V. Blomer, L. Goldmakher, B. Louvel, L-functions with n-th-order twists, IMRN 2014.
• L. Guth, J. Maynard, New large value estimates for Dirichlet polynomials, Ann. of Math. 203 (2026). https://doi.org/10.4007/annals.2026.203.2.6
• T. Tao, The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture, Algebra Number Theory 9 (2015).