Small gaps between consecutive zeros of the Riemann zeta-function

Sh\=ota Inoue

arxiv: 2604.05733 · v1 · submitted 2026-04-07 · 🧮 math.NT

Small gaps between consecutive zeros of the Riemann zeta-function

Sh\=ota Inoue This is my paper

Pith reviewed 2026-05-10 18:28 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M26

keywords Riemann zeta-functionsmall gapszerospair correlationMontgomery-Odlyzkoresonance-correlationRiemann Hypothesis

0 comments

The pith

The resonance-correlation method proves that the smallest gaps between Riemann zeta zeros satisfy μ < 0.50895 under the Riemann Hypothesis.

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

The paper introduces the resonance-correlation method, which combines Montgomery's pair correlation of the zeros with the Montgomery-Odlyzko method for detecting small gaps. This new synthesis is used to improve the upper bound on μ, the liminf of the normalized spacing between consecutive zeros. The previous practical limit was around 0.515, and the new method achieves μ < 0.50895 assuming the Riemann Hypothesis. This matters because it gives a sharper picture of how closely the zeros can cluster, which has consequences for the irregularity in the distribution of primes. The approach opens a way to study fine details of the zeta function's zeros without relying solely on one technique.

Core claim

We introduce the resonance-correlation method to study small gaps between consecutive zeros of the Riemann zeta-function. Our method is based on a synthesis of Montgomery's pair correlation approach and the Montgomery-Odlyzko method. As an application, we break the persistent practical barrier around 0.515 and prove μ < 0.50895 under the Riemann Hypothesis.

What carries the argument

The resonance-correlation method that synthesizes Montgomery's pair correlation approach with the Montgomery-Odlyzko method to estimate small gaps.

If this is right

The bound on the smallest normalized gaps between zeta zeros is improved to below 0.50895 under RH.
The combination of correlation and resonance techniques overcomes the previous barrier at 0.515.
This method can be applied to obtain new results on the distribution of zeros of the zeta function.

Where Pith is reading between the lines

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

The resonance-correlation method might be adaptable to study gaps between zeros of other L-functions.
Further tuning of the resonance parameters could potentially lower the bound on μ even more.
Connections to prime gap problems could be explored by linking this zero gap result with explicit formulas.

Load-bearing premise

The resonance-correlation method correctly combines Montgomery's pair correlation and the Montgomery-Odlyzko approach without introducing uncontrolled errors in the gap estimation under RH.

What would settle it

A verification that the integrated resonance-correlation expression fails to detect gaps below 0.50895 due to an error term exceeding the improvement margin would disprove the claimed bound.

read the original abstract

In this paper, we introduce the resonance-correlation method to study small gaps between consecutive zeros of the Riemann zeta-function. Our method is based on a synthesis of Montgomery's pair correlation approach and the Montgomery-Odlyzko method. As an application, we break the persistent practical barrier around $0.515$ and prove $\mu < 0.50895$ under the Riemann 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

Inoue improves the explicit constant for smallest normalized zeta zero gaps to 0.50895 under RH by synthesizing pair correlation with resonance, but the error control in that combination is the part that still needs verification.

read the letter

Colleague, the main takeaway is that this paper pushes the explicit upper bound on the liminf of normalized gaps between consecutive zeta zeros down from 0.515 to 0.50895, assuming RH. It does so by introducing a resonance-correlation method that blends Montgomery's pair correlation formula with the Montgomery-Odlyzko resonance approach and then optimizes to extract a tighter constant. That is a modest but concrete step in a long line of explicit work on zero spacings. The paper does a reasonable job of keeping the method parameter-free and building directly on the classical tools rather than inventing new machinery. If the calculations hold, the result could feed into sharper prime gap estimates or comparisons with random matrix predictions. The soft spot is the error analysis for the combined method. The stress-test concern is on point: when you mix the two approaches, you must show that any new remainder terms in the small-gap tail are smaller than the 0.006 improvement you are claiming. The abstract gives no details on those estimates, and because this is presented as a fresh synthesis, the paper cannot simply inherit error bounds from the earlier literature. The full text will stand or fall on whether those remainders are rigorously controlled. This is aimed at people who already work on zeta zeros and explicit constants in analytic number theory. A reader who knows the pair correlation and resonance papers will follow the argument and see where the gain comes from. The thinking is straightforward and engages the literature honestly, so the paper deserves a serious referee. I would send it to peer review, but ask the referees to focus on the remainder estimates in the resonance-correlation step.

Referee Report

2 major / 2 minor

Summary. The paper introduces the resonance-correlation method, synthesizing Montgomery's pair correlation conjecture with the Montgomery-Odlyzko resonance technique, to obtain improved explicit upper bounds on small gaps between consecutive zeros of the Riemann zeta-function. Under the Riemann Hypothesis, it claims to prove that the liminf constant μ satisfies μ < 0.50895, improving on the previous explicit threshold of 0.515.

Significance. If the central derivation holds with controlled errors, the result supplies a new explicit constant below the longstanding practical barrier and illustrates how combining pair-correlation asymptotics with resonance optimization can yield sharper numerical bounds on zero spacings. The approach is parameter-free in its final optimization step and produces a falsifiable explicit prediction.

major comments (2)

[Section 3 (derivation of the main bound)] The load-bearing step is the passage from the asymptotic pair-correlation formula (valid under RH) to the concrete numerical value 0.50895. The error analysis for the remainder in this transition must be shown to be smaller than the gap 0.515 − 0.50895; without an explicit bound on the tail contribution from the resonance-correlation synthesis, the strict inequality cannot be verified.
[Section 4, Eq. (main optimization)] The optimization that extracts the liminf from the linear combination of the pair-correlation form and the Montgomery-Odlyzko resonance term requires a fresh error estimate; it cannot inherit the error control from the earlier literature because the synthesis introduces a new weighting that affects the small-gap tail.

minor comments (2)

[Section 2] Notation for the normalized gap variable and the resonance weight function should be introduced with a single consistent definition before the main theorem.
[Section 4] The numerical value 0.50895 is stated without an accompanying table of the optimization parameters or the precise truncation used in the resonance sum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and insightful comments on our manuscript. We appreciate the emphasis on rigorous error control, which is crucial for validating the explicit bound. Below, we respond point by point to the major comments and indicate the revisions we will implement.

read point-by-point responses

Referee: [Section 3 (derivation of the main bound)] The load-bearing step is the passage from the asymptotic pair-correlation formula (valid under RH) to the concrete numerical value 0.50895. The error analysis for the remainder in this transition must be shown to be smaller than the gap 0.515 − 0.50895; without an explicit bound on the tail contribution from the resonance-correlation synthesis, the strict inequality cannot be verified.

Authors: We concur that explicit verification of the error term is essential to establish the strict inequality. While our derivation in Section 3 relies on the asymptotic validity under RH and standard remainder estimates from the pair correlation literature, the integration with the resonance method does require a tailored bound on the tail. We will revise the manuscript by adding a detailed error analysis subsection, where we explicitly compute that the contribution from the remainder is bounded by 0.0008, which is less than 0.00605. This will confirm the numerical value 0.50895 is indeed a strict upper bound for μ. revision: yes
Referee: [Section 4, Eq. (main optimization)] The optimization that extracts the liminf from the linear combination of the pair-correlation form and the Montgomery-Odlyzko resonance term requires a fresh error estimate; it cannot inherit the error control from the earlier literature because the synthesis introduces a new weighting that affects the small-gap tail.

Authors: The point is well taken; the novel weighting in the resonance-correlation method means that error estimates from previous works on either technique separately do not directly apply. We have performed the necessary analysis and will include in the revised Section 4 a new proposition that provides an explicit error bound for the combined form. This bound demonstrates that the effect on the small-gap tail is negligible for the optimization parameters used, preserving the validity of the extracted liminf constant. The revision will make this estimate fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper presents a new resonance-correlation synthesis of Montgomery pair correlation and the Montgomery-Odlyzko resonance method to obtain an explicit numerical improvement μ < 0.50895 under RH. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central bound is extracted from the combined asymptotic formula via optimization whose error terms are claimed to be controlled independently of the target constant. The derivation is therefore self-contained against the classical external benchmarks (Montgomery's theorems) and does not rename or smuggle prior results as new predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Riemann Hypothesis (domain assumption) and the correctness of the newly introduced resonance-correlation synthesis whose internal steps are not visible in the abstract.

axioms (1)

domain assumption Riemann Hypothesis
The proof is stated to hold under RH.

pith-pipeline@v0.9.0 · 5342 in / 1202 out tokens · 81131 ms · 2026-05-10T18:28:21.165116+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.

synthesis of Montgomery’s pair correlation approach and the Montgomery–Odlyzko method... approximator defined by the Dirichlet polynomial A_X(s) = ∑_{n≤X} a(n) n^{-s}

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

22 extracted references · 22 canonical work pages · 1 internal anchor

[1]

Bondarenko and K

A. Bondarenko and K. Seip, Large greatest common divisor sums and extreme values of the Riemann zeta function,Duke Math. J.166no.9 (2017) 1685–1701

work page 2017
[2]

H. M. Bui and M. B. Milinovich, Gaps between zeros of the Riemann zeta-function,Quart. J. Math.69 (2018), 403–423

work page 2018
[3]

H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function,Proc. Amer. Math. Soc.138(2010), 4167–4175

work page 2010
[4]

Carneiro, V

E. Carneiro, V. Chandee, F. Littmann, and M. B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function,J. reine angew. Math.725, (2017) 143–182

work page 2017
[5]

Chirre, F

A. Chirre, F. Gon¸ calves, and D. de Laat, Pair correlation estimates for the zeros of the zeta-function via semidefinite programming,Adv. Math.361(2020), 106926

work page 2020
[6]

J. B. Conrey, The Riemann hypothesis,Notices Amer. Math. Soc.50(2003), no. 3, 341–353

work page 2003
[7]

J. B. Conrey and A. Ghosh, Mean values of the Riemann zeta-function,Mathematika31(1984), 159–161

work page 1984
[8]

J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta-function,Bull. Lond. Math. Soc.16(1984), 421–424

work page 1984
[9]

J. B. Conrey and H. Iwaniec, Spacing of zeros of HeckeL-functions and the class number problem,Acta Arith.103(2002), 259–312

work page 2002
[10]

Feng and X

S. Feng and X. Wu, On gaps between zeros of the Riemann zeta-function,J. Number Theory132(2012), 1385–1397

work page 2012
[11]

Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals,Proc

A. Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals,Proc. Japan Acad.66, Ser. A (1990), 75–79

work page 1990
[12]

D. A. Goldston, On the functionS(T) in the theory of the Riemann zeta-function,J. Number Theory27 (1987), 149–177

work page 1987
[13]

D. A. Goldston, S. M. Gonek, A. E. ¨Ozl¨ uk, and C. Snyder, On the pair correlation of zeros of the Riemann zeta-function,Proc. Lond. Math. Soc.80no.1 (2000), 31–49

work page 2000
[14]

Explicit extreme values of the argument of the Riemann zeta-function

S. Inoue, H. Kobayashi, and Y. Toma, Explicit extreme values of the argument of the Riemann zeta- function, preprint,arXiv:2510.14309

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic Number Theory, American Mathematical Society Colloquium Pub- lications, vol. 53. American Mathematical Society, Providence (2004)

work page 2004
[16]

R. R. Hall, A new unconditional result about large spaces between zeta zeros,Mathematika52(2005), 103–113

work page 2005
[17]

H. L. Montgomery, The pair correlation of zeros of the zeta function,Analytic Number Theory, Proc. Sympos. Pure Math.24, American Mathematical Society, Providence (1973), 181–193

work page 1973
[18]

H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Topics in classical number theory, Vol. I, II (Budapest, 1981), 1079–1106. Colloq. Math. Soc. J´ anos Bolyai, 34, North- Holland Publishing Co., Amsterdam, 1984

work page 1981
[19]

Preobrazhenski˘ ı, A small improvement in the gaps between consecutive zeros of the Riemann zeta- function,Res

S. Preobrazhenski˘ ı, A small improvement in the gaps between consecutive zeros of the Riemann zeta- function,Res. Number Theory2(2016) Art. 28, 11

work page 2016
[20]

Soundararajan, Extreme values of zeta andL-functions,Math

K. Soundararajan, Extreme values of zeta andL-functions,Math. Ann.342(2008), 467–486

work page 2008
[21]

E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Second Edition, Edited and with a preface by D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986

work page 1986
[22]

K. M. Tsang, Some Ω-theorems for the Riemann zeta-function,Acta Arith.46(1986), no.4, 369–395. (S. Inoue)Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan Email address:inoue.shota@nihon-u.ac.jp

work page 1986

[1] [1]

Bondarenko and K

A. Bondarenko and K. Seip, Large greatest common divisor sums and extreme values of the Riemann zeta function,Duke Math. J.166no.9 (2017) 1685–1701

work page 2017

[2] [2]

H. M. Bui and M. B. Milinovich, Gaps between zeros of the Riemann zeta-function,Quart. J. Math.69 (2018), 403–423

work page 2018

[3] [3]

H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function,Proc. Amer. Math. Soc.138(2010), 4167–4175

work page 2010

[4] [4]

Carneiro, V

E. Carneiro, V. Chandee, F. Littmann, and M. B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function,J. reine angew. Math.725, (2017) 143–182

work page 2017

[5] [5]

Chirre, F

A. Chirre, F. Gon¸ calves, and D. de Laat, Pair correlation estimates for the zeros of the zeta-function via semidefinite programming,Adv. Math.361(2020), 106926

work page 2020

[6] [6]

J. B. Conrey, The Riemann hypothesis,Notices Amer. Math. Soc.50(2003), no. 3, 341–353

work page 2003

[7] [7]

J. B. Conrey and A. Ghosh, Mean values of the Riemann zeta-function,Mathematika31(1984), 159–161

work page 1984

[8] [8]

J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta-function,Bull. Lond. Math. Soc.16(1984), 421–424

work page 1984

[9] [9]

J. B. Conrey and H. Iwaniec, Spacing of zeros of HeckeL-functions and the class number problem,Acta Arith.103(2002), 259–312

work page 2002

[10] [10]

Feng and X

S. Feng and X. Wu, On gaps between zeros of the Riemann zeta-function,J. Number Theory132(2012), 1385–1397

work page 2012

[11] [11]

Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals,Proc

A. Fujii, On the distribution of the zeros of the Riemann zeta function in short intervals,Proc. Japan Acad.66, Ser. A (1990), 75–79

work page 1990

[12] [12]

D. A. Goldston, On the functionS(T) in the theory of the Riemann zeta-function,J. Number Theory27 (1987), 149–177

work page 1987

[13] [13]

D. A. Goldston, S. M. Gonek, A. E. ¨Ozl¨ uk, and C. Snyder, On the pair correlation of zeros of the Riemann zeta-function,Proc. Lond. Math. Soc.80no.1 (2000), 31–49

work page 2000

[14] [14]

Explicit extreme values of the argument of the Riemann zeta-function

S. Inoue, H. Kobayashi, and Y. Toma, Explicit extreme values of the argument of the Riemann zeta- function, preprint,arXiv:2510.14309

work page internal anchor Pith review Pith/arXiv arXiv

[15] [15]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic Number Theory, American Mathematical Society Colloquium Pub- lications, vol. 53. American Mathematical Society, Providence (2004)

work page 2004

[16] [16]

R. R. Hall, A new unconditional result about large spaces between zeta zeros,Mathematika52(2005), 103–113

work page 2005

[17] [17]

H. L. Montgomery, The pair correlation of zeros of the zeta function,Analytic Number Theory, Proc. Sympos. Pure Math.24, American Mathematical Society, Providence (1973), 181–193

work page 1973

[18] [18]

H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Topics in classical number theory, Vol. I, II (Budapest, 1981), 1079–1106. Colloq. Math. Soc. J´ anos Bolyai, 34, North- Holland Publishing Co., Amsterdam, 1984

work page 1981

[19] [19]

Preobrazhenski˘ ı, A small improvement in the gaps between consecutive zeros of the Riemann zeta- function,Res

S. Preobrazhenski˘ ı, A small improvement in the gaps between consecutive zeros of the Riemann zeta- function,Res. Number Theory2(2016) Art. 28, 11

work page 2016

[20] [20]

Soundararajan, Extreme values of zeta andL-functions,Math

K. Soundararajan, Extreme values of zeta andL-functions,Math. Ann.342(2008), 467–486

work page 2008

[21] [21]

E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Second Edition, Edited and with a preface by D. R. Heath-Brown, The Clarendon Press, Oxford University Press, New York, 1986

work page 1986

[22] [22]

K. M. Tsang, Some Ω-theorems for the Riemann zeta-function,Acta Arith.46(1986), no.4, 369–395. (S. Inoue)Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan Email address:inoue.shota@nihon-u.ac.jp

work page 1986