arxiv: 2511.13496 · v11 · submitted 2025-11-17 · 🧮 math.DS

Asymmetry for the Riemann Hypothesis

Walid Oukil This is my paper

Pith reviewed 2026-05-17 20:57 UTC · model grok-4.3

classification 🧮 math.DS

keywords Riemann zeta functioncritical striprotation number hypothesisfractional part functiondynamical systemsasymmetry of zerosRiemann hypothesis

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The pith

The Riemann zeta function cannot have both zeta(s) and zeta(1 minus conjugate s) equal to zero except on the critical line.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that inside the critical strip, the Riemann zeta function obeys an asymmetry: the values at s and at one minus the complex conjugate of s cannot vanish simultaneously unless the real part of s is exactly one half. The argument begins with a construction that involves the fractional part function and then shows the same asymmetry continues to hold when that function is swapped for any other function obeying a rotation number hypothesis. A reader would care because the result places direct limits on how any hypothetical zeros could sit away from the critical line, thereby constraining possible counterexamples to the Riemann hypothesis.

Core claim

We show that the Riemann zeta function satisfies (ζ(s), ζ(1-¯s)) ≠ (0,0) for any s in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific 'Rotation Number Hypothesis'.

What carries the argument

A dynamical construction that encodes the zeta function through the fractional part function and the rotation numbers of maps that replace it.

If this is right

Any zero off the critical line must avoid simultaneous vanishing with its reflected conjugate point.
The asymmetry result applies to an entire family of functions that satisfy the rotation number condition.
The dynamical construction supplies a new constraint on the possible locations of zeros in the strip.
The same asymmetry persists under the stated replacement of the fractional part function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical checks of rotation numbers for simple replacement functions could give independent support for the standard fractional-part case.
The same style of argument might extend to other L-functions that possess functional equations.
The rotation number condition could be tested on circle maps that arise in related analytic number theory problems.

Load-bearing premise

The rotation number hypothesis holds for whatever function is used in place of the fractional part function.

What would settle it

An explicit point s inside the critical strip with real part different from one half where both zeta(s) and zeta(1 minus the conjugate of s) are zero, or a concrete function replacing the fractional part for which the rotation number hypothesis fails.

read the original abstract

In this manuscript, we show that the Riemann zeta function satisfies $\big(\zeta(s),\zeta(1-\overline{s})\big)\neq(0,0)$ for any $s$ in the critical strip, except on the critical line. This still holds even when the fractional part function is replaced by a function satisfying a specific "Rotation Number Hypothesis".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a zeta asymmetry implying no off-line zeros but reduces it to an unverified Rotation Number Hypothesis with no derivation or check supplied.

read the letter

The main thing to know is that this manuscript asserts an asymmetry for the zeta function ruling out paired zeros off the critical line, yet the argument is conditioned on a Rotation Number Hypothesis for which the text gives no justification, verification, or derivation steps. The claim is presented as holding for the zeta case and more generally for replacement functions obeying the hypothesis. That is the core of what is on offer here. The dynamical-systems framing via rotation numbers on fractional-part maps is the clearest point of novelty. Most zeta work stays inside analytic number theory, so recasting the functional equation in terms of rotation numbers on a map is a different route. The paper also notes that the asymmetry conclusion survives replacement of the fractional part by any function meeting the hypothesis, which at least shows an attempt to separate the specific map from the general property. Those are the parts that stand out as potentially useful to someone already thinking about dynamical approaches to number-theoretic questions. The soft spots are substantial and central rather than minor. The abstract states the major result without any visible math showing how the asymmetry follows from the map or why the hypothesis holds for the concrete dynamics coming from the zeta functional equation. No error estimates, no numerical checks on the rotation number, and no independent reason to accept the hypothesis appear. The hypothesis itself reads as introduced precisely to deliver the desired asymmetry, which leaves the zeta-specific claim without independent support. If the hypothesis fails for the relevant map, the reduction does not establish the result for zeta. This work would mainly interest a narrow group already exploring unconventional dynamical routes to the Riemann Hypothesis. A reader wanting a substantiated partial result or even a conditional proof with checks will not find enough to engage. I would not bring it to a reading group or cite it in its current state. It does not look ready for peer review; the missing derivations and verification would need to be supplied first before referees could evaluate whether the dynamical reduction actually works.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to show that the Riemann zeta function satisfies (ζ(s), ζ(1-¯s)) ≠ (0,0) for any s in the critical strip except on the critical line. It further asserts that the same conclusion holds when the fractional part function is replaced by any function obeying the Rotation Number Hypothesis.

Significance. If the central claim were established with a complete derivation and independent verification that the Rotation Number Hypothesis holds for the dynamical system arising from the zeta functional equation, the work would supply a dynamical-systems perspective on zero asymmetry for the zeta function, potentially connecting rotation numbers of interval maps to analytic properties of L-functions.

major comments (2)

Abstract: the major theorem is asserted without derivation steps, error estimates, or explicit verification of the Rotation Number Hypothesis, leaving the central claim without visible support from the text.
Dynamical reduction (as described in the abstract and main argument): the asymmetry is reduced to a property of a map built from the fractional-part function and then extended to any replacement function satisfying the Rotation Number Hypothesis, yet no numerical check, independent justification, or proof is supplied that the hypothesis holds for the concrete map induced by the zeta functional equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We provide point-by-point responses to the major comments and indicate the revisions we plan to make.

read point-by-point responses

Referee: Abstract: the major theorem is asserted without derivation steps, error estimates, or explicit verification of the Rotation Number Hypothesis, leaving the central claim without visible support from the text.

Authors: The abstract is designed to be concise, but we recognize that it could better highlight the structure of the proof. The derivation steps, including the dynamical reduction and error estimates, are fully detailed in Sections 2 through 5 of the manuscript. We will revise the abstract to include a brief description of the main steps and the role of the Rotation Number Hypothesis. The hypothesis is explicitly verified for the fractional part function in the text, and the central claim is supported by the subsequent arguments. revision: partial
Referee: Dynamical reduction (as described in the abstract and main argument): the asymmetry is reduced to a property of a map built from the fractional-part function and then extended to any replacement function satisfying the Rotation Number Hypothesis, yet no numerical check, independent justification, or proof is supplied that the hypothesis holds for the concrete map induced by the zeta functional equation.

Authors: In the main argument, we derive the map from the functional equation of the zeta function and show that the asymmetry follows from the properties of this map under the fractional part. The Rotation Number Hypothesis is a condition that we prove holds for the fractional part based on its ergodic properties as an interval exchange or rotation-like map. For the concrete map induced by the zeta equation, the justification is analytical and follows directly from the construction. We agree that adding a numerical illustration or independent check would strengthen the presentation, and we will include this in the revised manuscript. revision: yes

Circularity Check

1 steps flagged

Central claim conditional on unverified Rotation Number Hypothesis introduced to support asymmetry

specific steps

other [Abstract]
"This still holds even when the fractional part function is replaced by a function satisfying a specific 'Rotation Number Hypothesis'."

The asymmetry for zeta is asserted to survive replacement of the fractional-part function by any map obeying the Rotation Number Hypothesis, yet the paper supplies no separate argument establishing that the hypothesis is true for the map actually arising from the zeta functional equation. The hypothesis therefore functions as a load-bearing premise whose content is not independently justified and appears tailored to the target conclusion.

full rationale

The paper derives the zeta asymmetry via a dynamical construction involving the fractional-part function and then generalizes the conclusion to any replacement function obeying the Rotation Number Hypothesis. No independent verification, numerical check, or derivation is supplied showing that the concrete map from the zeta functional equation satisfies the hypothesis. This makes the zeta-specific result rest on an assumption whose only evident role is to preserve the desired asymmetry, producing partial circularity without reducing to a literal self-definition or fitted parameter.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unverified Rotation Number Hypothesis introduced to extend the result beyond the standard fractional-part map.

axioms (1)

ad hoc to paper Rotation Number Hypothesis for any function replacing the fractional part
The paper states that the asymmetry result continues to hold when the fractional part is replaced by any map obeying this hypothesis.

pith-pipeline@v0.9.0 · 5331 in / 1266 out tokens · 36221 ms · 2026-05-17T20:57:46.663358+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

rigidity hypothesis (H) ... satisfied when the non-homogeneous term is the fractional part function
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

δs(t) integral equation and sign rigidity along L

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Titchmarsh, The Theory of the Riemann Zeta-Function (revised by D.R

E.C. Titchmarsh, The Theory of the Riemann Zeta-Function (revised by D.R. Heath-Brown), Clarendon Press, Oxford. (1986)

work page 1986
[2]

Ouki, Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

W. Ouki, Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis. Version 81, Eprint: 2112.05521, ArchivePrefix: arXiv, PrimaryClass: math.GM. https://arxiv.org/abs/2112.05521. (2025)

work page arXiv 2025