# The quasi-Riemann hypothesis

> No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-003/
Last updated: 2026-10-11

**No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8.**

**Trust:** Formal proof listed · **Papers:** 3 · **First paper:** September 30, 2026 · **Field:** Number theory · **Entry:** 003

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Riemann's zeta function and its relatives, the Dirichlet L-functions, have zeros that control how primes are spread out. The quasi-Riemann hypothesis asks for one fixed wall below 1 with no zeros to its right. This paper proves such a wall at 7/8 (Lean proof listed). It is not the Riemann hypothesis, which needs every zero at real part exactly 1/2. Two companion papers add a separate wall at 11/12 and a uniform gap for Landau–Siegel zeros.

## At a glance

- **First step:** 1896 (Hadamard and de la Vallée Poussin, no zeros on the line Re s = 1)
- **Significance:** Landmark. Every zero-free zone known before narrowed with height. This gives a fixed one and rules out Landau–Siegel zeros above 7/8.
- **Trust:** Formal proof listed
- **Field:** Number theory (3 papers)

**Fixed wall with no zeros of ζ to its right**

| Year | Who | Value |
| --- | --- | --- |
| 1896 | Hadamard; de la Vallée Poussin (no zeros on Re s = 1) | 1 |
| 2026 | 11/12 (companion paper) | 0.9167 (this result) |
| 2026 | 7/8 (this paper) | 0.875 (this result) |

Lower is better. Before 2026 no fixed wall below 1 was known; 1/2 is the most the Riemann hypothesis would give.

## The explanation

Add 1 + 1/4 + 1/9 + 1/16 and keep going, using one over every square number. The total settles at π²/6, about 1.6449. Replace the power 2 by any number s and you get Riemann's zeta function, written ζ(s). A standard extension lets s be a complex number, with a real part and an imaginary part. Call the imaginary part the height. The first zero of ζ, a point where it equals 0, has real part 1/2 and height about 14.1347. The next two also have real part 1/2, at heights about 21.0220 and 25.0109.

The real part of each zero is what matters here. The zeros of interest all have real part between 0 and 1, a band called the critical strip. The Riemann hypothesis says every one of them has real part exactly 1/2. It is still open, and this work does not change that[1]. The quasi-Riemann hypothesis is a much weaker wish. It asks for one fixed wall below 1, such as 0.99. No zero may lie to the right of that wall, at any height. It says nothing about where the zeros sit, only that they keep a fixed distance from 1.

Zeros near real part 1 matter. In 1859, Riemann tied the zeros of ζ to how primes are spread out. In 1896, Hadamard and de la Vallée Poussin showed that no zero has real part exactly 1. That gave the prime number theorem, which counts primes up to a given size. Later work cleared a thin zone just left of the line at real part 1. But every such zone narrows as the height grows. In 1958, Vinogradov and Korobov found a different zone, and it narrows too. No fixed wall was known.

The same question covers a wider family. Take 1 − 1/3 + 1/5 − 1/7 and so on. Each odd number gets + or − by its remainder after dividing by 4. The total is π/4, about 0.7854. That sign pattern is a Dirichlet character: a rule that repeats with some period and respects multiplication. Each character gives its own Dirichlet L-function, and ζ is the case where every sign is +. Dirichlet built these in 1837 to find primes in arithmetic progressions, such as 1, 5, 9, 13 and so on. For them the classical zones narrow as the height or the period grows. They also allow one possible exception. That is a single real zero very close to 1. It can occur only for a character with values 0, 1 and −1. Such a zero is called a Landau–Siegel zero, and nobody could rule it out.

The first paper, dated 30 September 2026, proves that the wall exists and puts it at 7/8, or 0.875[2]. No Dirichlet L-function, ζ included, has a zero with real part above 7/8. The same wall holds for every character, every period and every height[3]. The line at 7/8 itself is not covered. Any Landau–Siegel zero above 7/8 is ruled out as well[4]. For the basic, so-called primitive L-functions, a mirror symmetry then keeps every such zero between 1/8 and 7/8. That still allows zeros at real part 0.6 or 0.8, which the Riemann hypothesis forbids.

The one obstacle was that classical methods lose strength as the height and the period grow. A fixed wall needs an argument whose margin is the same for every function. The paper gets this by working in a larger number system, the Eisenstein integers. These are numbers a + bω, with a and b whole numbers. Here ω is a cube root of 1 other than 1. They are the whole numbers of the number field Q(√−3). The paper proves the wall for a whole family of functions over that field, the finite-order Hecke L-functions[2]. The whole family is needed even for ζ, because the method keeps producing twisted relatives of the function under study[5]. It draws on a cubic version of the theta series, from S. J. Patterson in 1977[6].

The core move bounds one carefully weighted sum directly. It also compares that sum with an integral containing 1 divided by the L-function[7]. If some zero lay past the wall, take the rightmost possible zeros. The two computations would show that this reciprocal stays well behaved a fixed distance left of them. A reciprocal cannot stay well behaved at a zero, so no such zero exists. A first pass gives a wall at 11/12, about 0.9167, and a refined pass reaches 7/8[8]. Each Dirichlet L-function, times a partner, matches a Hecke L-function up to finitely many harmless factors[9]. So the wall passes down to every Dirichlet L-function.

Two companion papers sit beside it. One, dated 5 October 2026, claims to prove the 11/12 wall by itself, as a shorter intermediate step[10]. The other, dated 1 October 2026, takes a separate, mostly algebraic route aimed only at Landau–Siegel zeros[11]. It proves that one constant c > 0 works for every such character with period q of at least 3[12]. Each real zero β of such a function then satisfies (1 − β) × log q ≥ c. It gives no value for c. The first paper also draws consequences. One concerns the smallest remainder that is not a perfect square, modulo a prime. Modulo 7, squares leave remainders 1, 2 and 4, so that smallest non-square is 3. The paper claims this number is at most a fixed power of log p for every odd prime p[13].

How sure is this? A computer proof checker called Lean has checked formal versions of the 7/8 wall. They cover ζ and every Dirichlet L-function, with one wall for all of them. They also cover the finite-order Hecke L-functions over Q(√−3). Lean has also checked the uniform gap for real zeros from the 1 October paper, with no explicit c. The paper's later applications, such as the non-square bound, are outside that check. The 11/12 companion is not in the Lean scope note. No specialist review has been published.

**What it doesn't show.** This does not prove the Riemann hypothesis: zeros may still lie anywhere from 1/8 to 7/8, not only at 1/2.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Landmark if it holds. Lean checks formal versions of the 7/8 wall and the real-zero gap. No specialist review is published, the applications are unchecked, and the Riemann hypothesis stays open.

## Words used

- **Riemann zeta function:** The function ζ(s) = 1 + 1/2^s + 1/3^s + …, extended to complex numbers s.
- **Zero:** A complex number where the function equals 0.
- **Riemann hypothesis:** The open conjecture that every zero in the critical strip has real part exactly 1/2.
- **Quasi-Riemann hypothesis:** The weaker claim that some fixed number below 1 has no zeros to its right, at any height.
- **Dirichlet character:** A repeating rule assigning values to whole numbers that respects multiplication, used to sort primes by remainder.
- **Dirichlet L-function:** The sum of a Dirichlet character's value at n divided by n^s, over all n; ζ is the simplest one.
- **Landau–Siegel zero:** A possible single real zero very close to 1, for a character with values 0, 1 and −1.
- **Hecke L-function:** An analogue of a Dirichlet L-function built over a larger number system, here the field Q(√−3). "Finite-order" means some fixed power of its character equals 1 everywhere.

## The original papers (3)

- [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf) - September 30, 2026 · Formal proof listed
- [The Quasi-Riemann Hypothesis (alternate 11/12 proof)](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf) - October 5, 2026
- [Uniform exclusion of Landau–Siegel zeros](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf) - October 1, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/003.md)

## Quoted lines from the paper (13)

[1] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The theorem does not establish the Riemann hypothesis or its generalized versions, which place nontrivial zeros on $\Re s=1/2$.

[2] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The same holds for every Dirichlet $L$-function, including $\zeta(s)$.

[3] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The problem here is uniform exclusion: to find a constant $\sigma_0<1$, independent of the character, its conductor, and the height, such that these functions have no zeros in $\Re s>\sigma_0$.

[4] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The boundary $\Re s=7/8$ is not included, while any exceptional real zero in $(7/8,1)$ is excluded.

[5] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The entire family is needed even for the consequence about $\zeta$, because the Poisson rows introduce finite-order Hecke twists of the target.

[6] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> Kubota's metaplectic theory and Patterson's cubic theta series provide the automorphic setting.

[7] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> A power saving in both comparisons then continues that reciprocal across the rightmost possible zeros.

[8] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> Part … starts from that conclusion and introduces prime compensation, asymmetric scales, and two additional moment estimates.

[9] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> The omitted factors are nonzero in $\Re s>0$, and the principal pole is handled separately.

[10] From [The Quasi-Riemann Hypothesis (alternate 11/12 proof)](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf):

> Theorem … is a natural intermediate step toward the stronger zero-free region $\Rea s>7/8$ established in.

[11] From [Uniform exclusion of Landau–Siegel zeros](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf):

> The remaining argument is algebraic.

[12] From [Uniform exclusion of Landau–Siegel zeros](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf):

> 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 $(1-\beta)\log q\ge c$.

[13] From [The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf):

> There are absolute constants $A,C>0$ such that, for every odd prime $p$, the least positive quadratic nonresidue $n(p)$ satisfies \[ n(p)\le C(\log p)^A. \]

## Other sources

- [J. Hadamard, “Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques”, Bull. Soc. Math. France 24 (1896), 199–220](https://doi.org/10.24033/bsmf.545)
- [C.-J. de la Vallée Poussin, “Recherches analytiques sur la théorie des nombres premiers”, Ann. Soc. Sci. Bruxelles 20 (1896), 183–256](https://books.google.com/books?id=7e0GAAAAYAAJ)
- [B. Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (1859)](https://www.claymath.org/library/historical/riemann/r.html)
- [P. G. L. Dirichlet (1837), on primes in arithmetic progressions; English translation by R. Stephan, arXiv:0808.1408](https://arxiv.org/abs/0808.1408)
- [I. M. Vinogradov, “A new estimate of the function ζ(1+it)”, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 161–164](https://www.mathnet.ru/eng/im3962)
- [N. M. Korobov, “Estimates of trigonometric sums and their applications”, Uspekhi Mat. Nauk 13 (1958), 185–192](https://www.mathnet.ru/eng/rm7458)
- [S. J. Patterson, “A cubic analogue of the theta series”, J. reine angew. Math. 296 (1977), 125–161](https://doi.org/10.1515/crll.1977.296.125)
- [Clay Mathematics Institute, Riemann Hypothesis, Millennium Prize Problems](https://www.claymath.org/millennium/riemann-hypothesis/)
- [T. Tao, “The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture”, Algebra & Number Theory 9 (2015), 1005–1034](https://doi.org/10.2140/ant.2015.9.1005)
- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

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