The quasi-Riemann hypothesis

1 · The essentials

These works study where certain functions in number theory take the value zero. The Riemann zeta function is among the functions considered. The main result excludes every zero when the real part of the input is strictly greater than $7/8$ . It applies to all Dirichlet L-functions and to another family specified below. The point 1 may still be a pole, rather than a place where the function has an ordinary value. The manuscript presents this result as resolving the quasi-Riemann hypothesis, and the main result has a Lean formalization. Two companion texts give, respectively, a different exclusion covering a smaller region and a uniform constraint on certain real zeros.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect picture, imagine a map on which you are looking for marks: the result guarantees that no marks lie to the right of a particular vertical line. Moving that line from $11/12$ to $7/8$ enlarges the region guaranteed to be clear; it does not describe the whole map.

Why it matters

The main result provides one shared zero-free region for every function in the two stated families, including the zeta function. The third manuscript formulates a uniform exclusion of Landau–Siegel zeros using a constant that works for every function it considers. These statements therefore concern entire families, rather than a few individual functions.

2 · Going further

The problem is to exclude zeros from a half-plane in the complex plane. The main manuscript states that every Dirichlet L-function and every finite-order Hecke L-function over $K = Q (- 3)$ is zero-free in $ℜ s > 7/8$ , with a principal pole at $s = 1$ allowed. The Riemann zeta function is explicitly included. The text presents this statement as resolving the quasi-Riemann hypothesis; the main result is identified as Lean-formalized.

The second manuscript gives an alternative argument, shown in the manuscript, for the half-plane $ℜ s > 11/12$ , covering the same families of functions. Since $7/8 < 11/12$ , the main result excludes zeros from a larger region. This companion also claims that its result rules out Landau–Siegel zeros. The abstracts describe neither the steps nor the technical idea of the arguments: they allow a comparison of the conclusions, but not an explanation of how the arguments work.

The third manuscript, also identified as Lean-formalized, states a uniform exclusion: there is an absolute constant $c > 0$ such that every real zero $β \in (0, 1)$ of every primitive nonprincipal real Dirichlet L-function of conductor $q \geq 3$ satisfies $(1 - β) lo g q \geq c$ . In other words, under these precise conditions, the distance from 1 is at least $c / lo g q$ . This text supplies a complementary quantitative formulation; the abstracts do not specify dependencies between its arguments and those of the other two manuscripts.

Useful notions

Complex numbers and real parts
Zeros and poles of a function
Half-planes in the complex plane
Logarithms and inequalities
The idea of a family of L-functions

What was known before / what is new

Before

The source names the quasi-Riemann hypothesis and Landau–Siegel zeros as the questions being addressed. It supplies no history, account of earlier results, or attribution of these problems. The text with threshold $11/12$ is presented as a companion offering a different argument, with no chronology specified.

New

The main announced contribution is the zero-free half-plane $ℜ s > 7/8$ for all Dirichlet L-functions and all finite-order Hecke L-functions over $Q (- 3)$ . One companion gives a different argument for the threshold $11/12$ . A third states the uniform exclusion $(1 - β) lo g q \geq c$ under the specified hypotheses.

Limits and assumptions

The half-planes use strict inequalities: these statements do not settle what happens on their boundaries or in regions to the left. For Hecke L-functions, the result covers only finite-order functions over the specified field; a principal pole at 1 remains allowed. The uniform inequality concerns only real zeros in (0,1) of primitive nonprincipal real Dirichlet L-functions with conductor at least 3. The main result and the third manuscript are identified as Lean-formalized, but the source does not detail the technical scope of those formalizations; the second manuscript is not identified as formalized. The abstracts give no list of remaining open questions.

Mini glossary

Zero: An input at which a function takes the value zero.
Real part: The real coordinate of a complex number, written here as $ℜ s$ .
Half-plane: A region on one side of a line, here specified by a condition such as $ℜ s > 7/8$ .
Pole: A singularity where a function becomes unbounded, distinct from a zero.
Absolute constant: In the uniform statement, a positive constant shared by all functions and zeros satisfying the hypotheses.

Keywords : Quasi-Riemann hypothesis · L-functions · Landau–Siegel zeros

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that every Dirichlet L-function, including $ζ (s)$ , is zero-free in $ℜ s > 7/8$ , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $Q (- 3)$ . A companion gives a different proof of the zero-free half-plane $ℜ s > 11/12$ .

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/003.md

Manuscripts in this family

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The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $ℜ s > 7/8$

30/09/2026

Verified by Lean

We prove that all finite-order Hecke L-functions over $Q (- 3)$ and all Dirichlet L-functions are zero-free in the half-plane $ℜ s > 7/8$ , 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.

PDF GitHub folder Lean formalization

The Quasi-Riemann Hypothesis (alternate 11/12 proof)

05/10/2026

To be verified

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 $Re s > 11/12$ . More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over $K = Q (- 3)$ . In particular, this rules out the existence of Landau–Siegel zeros.

PDF GitHub folder

Uniform exclusion of Landau–Siegel zeros

01/10/2026