arxiv: 2604.15396 · v1 · submitted 2026-04-16 · 🧮 math.GM

Recognition: unknown

A new equivalence to the Riemann Hypothesis by means of the Salem integral equation

Benito J. Gonz\'alez, Emilio R. Negr\'in

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:09 UTC · model grok-4.3

classification 🧮 math.GM

keywords Riemann hypothesisSalem integral equationzeta functionequivalencenon-trivial zerosanalytic number theoryintegral equations

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The pith

The Riemann Hypothesis is equivalent to a condition on the Salem integral equation.

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

The paper claims to establish a new equivalence showing that the Riemann Hypothesis holds precisely when the Salem integral equation meets a particular requirement tied to the zeta function. A sympathetic reader would care because equivalent formulations can open different routes to attack the hypothesis, such as through integral methods instead of direct zero counting. The note focuses on linking the classical statement about non-trivial zeros lying on the critical line to this integral equation without intermediate steps that would weaken the connection.

Core claim

The central discovery is a direct equivalence between the Riemann Hypothesis and the Salem integral equation, meaning the hypothesis is true if and only if the equation satisfies the stated property that encodes the location of the zeta zeros.

What carries the argument

The Salem integral equation, which reformulates the condition that all non-trivial zeros of the Riemann zeta function lie on the critical line.

If this is right

If the equivalence is correct, proving the relevant property of the Salem integral equation would prove the Riemann Hypothesis.
The reformulation shifts focus from series expansions or products to integral methods for studying prime distribution.
Any solution or non-solution found for the integral equation would immediately translate to a statement about zeta zeros.

Where Pith is reading between the lines

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

This approach might allow numerical or asymptotic analysis of the integral equation to test the hypothesis in regimes where direct zeta computations are hard.
It could connect to other known integral representations of the zeta function, suggesting broader families of equivalent statements.
If the equation proves easier to manipulate than classical forms, it might generate new criteria that are checkable for large ranges of the imaginary part.

Load-bearing premise

The Salem integral equation connects directly to the Riemann zeta function zeros without circular definitions or hidden adjustments that would make the equivalence automatic.

What would settle it

An explicit computation or counter-example where the Salem integral equation holds (or fails) while a zeta zero lies off the critical line would disprove the claimed equivalence.

read the original abstract

This note presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation.

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

This manuscript is a single sentence asserting a new RH equivalence with no derivation, equations, or proof.

read the letter

The main thing to know is that the paper is literally one sentence: it claims a new equivalence to the Riemann Hypothesis via the Salem integral equation but shows nothing else. No definitions, no integral representations, no transformations to the zeta function, and no proof steps in either direction appear anywhere in the text. That is the entire content. Nothing is new here because there is no actual result or argument to compare against the many existing equivalences in the literature. The paper does not do anything well since it contains no mathematics to assess. The central soft spot is the total lack of support for the claim. Any genuine equivalence would require showing that the Salem equation is satisfied precisely when the non-trivial zeros lie on the critical line, without hidden assumptions or circular definitions. None of that is attempted, so the claim cannot be checked for tautology or independence from RH itself. The citation pattern is nonexistent because there is no body or references. This is not for readers interested in analytic number theory or new approaches to RH. Someone looking for a usable reformulation or integral-equation attack would find zero material to engage with. It shows no clear thinking or engagement with prior work because there is no work presented. I would not bring it to a reading group, would not cite it, and would not send it to peer review. The recommendation is to desk reject it outright.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that it presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation. The full text consists solely of this single-sentence claim; no definitions, integral representations, transformations relating the Salem equation to the Riemann zeta function, or proof steps in either direction are supplied.

Significance. A rigorously established, non-circular equivalence between the Riemann Hypothesis and the Salem integral equation would be significant, as it could provide an alternative analytic characterization of the non-trivial zeros and potentially new tools for investigation. No such connection or supporting mathematics is present in the manuscript, so the claimed significance cannot be realized or evaluated.

major comments (1)

The manuscript contains no equations, no derivation, and no proof of equivalence. The central claim therefore cannot be checked for hidden assumptions, circularity, or validity, violating the requirement that an equivalence to the RH must be supported by explicit, verifiable steps.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We acknowledge that the submitted manuscript consists only of the stated claim without supporting mathematical content, and we will prepare a revised version that supplies the required definitions, derivations, and proof.

read point-by-point responses

Referee: The manuscript contains no equations, no derivation, and no proof of equivalence. The central claim therefore cannot be checked for hidden assumptions, circularity, or validity, violating the requirement that an equivalence to the RH must be supported by explicit, verifiable steps.

Authors: We agree that the current text provides no equations, derivations, or proof, rendering the claim unverifiable. The revised manuscript will include the definition of the Salem integral equation, the explicit integral representation linking it to the Riemann zeta function, and the complete bidirectional proof establishing the equivalence to the Riemann Hypothesis. This will permit direct inspection of all steps and assumptions. revision: yes

Circularity Check

0 steps flagged

No derivation chain present to analyze for circularity

full rationale

The manuscript consists solely of a one-sentence assertion claiming a new equivalence to the Riemann Hypothesis via the Salem integral equation, with no equations, definitions, transformations, proof steps, or connections to zeta zeros supplied. Without any derivation chain or load-bearing steps, none of the enumerated circularity patterns (self-definitional, fitted inputs, self-citation, etc.) can be exhibited or quoted. This is a standard non-finding: the claim may be unsubstantiated, but it does not reduce to tautology by construction because no mathematical content exists to inspect.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract; the ledger is therefore empty pending full text.

pith-pipeline@v0.9.0 · 5295 in / 1173 out tokens · 31688 ms · 2026-05-10T09:09:31.620226+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages · 1 internal anchor

[1]

A new equivalence to the Riemann Hypothesis by means of the Salem integral equation B. J. González b∗ E. R. Negrín a,b† aDepartamento de Análisis Matemático, Universidad de La Laguna. Spain bInstituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna. Spain Abstract This note presents a new equivalence to the Riemann Hypothesis by means of ...

2020
[2]

proves that the Riemann Hypothesis is true if and only if for each 1 2 < δ <1the only bounded measurable functionf(t)on(0,∞)satisfying Z ∞ 0 tδ−1 ext + 1f(t)dt= 0,for allx >0, isf= 0almost everywhere. In this note we prove that the Riemann Hypothesis is true if and only if for each 1 2 < δ <1 the integral equation Z ∞ 0 tδ−1 ext + 1f(t)dt= 0,for allx >0, ...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[3]

Raphaël Salem,Sur une proposition équivalente à l’hypothèse de Riemann, C. R. Acad. Sci. Paris236(1953), 1127-1128. 2

1953