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

Hirotaka Kobayashi; Sh\=ota Inoue; Yuichiro Toma

REVIEW 2 major objections 2 minor 1 cited by

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

T0 review · 2 major / 2 minor · reviewed 2026-05-18 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Explicit extreme values of the Riemann zeta-function argument in short intervals improve r-gap bounds between zeros.

desk verdict They turn asymptotic large-value results for arg(ζ) into explicit short-interval bounds and use them to improve the constant in the Conrey–Turnage-Butterbaugh r-gap theorem. read the letter →

arxiv 2510.14309 v2 submitted 2025-10-16 math.NT

Shōta Inoue , Hirotaka Kobayashi , Yuichiro Toma This is my paper · ORCID

classification math.NT MSC 11M06

keywords Riemannzeta-functionargumentextremevaluesshortintervalsr-gapszeros

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The authors derive explicit upper and lower bounds on the size of the argument of the Riemann zeta-function inside short intervals on the critical line. These bounds are fed into existing zero-counting machinery to produce a quantitative improvement over the Conrey–Turnage-Butterbaugh theorem on the minimal spacing of r consecutive zeros. A reader would care because sharper control of the zeta-function's phase in small regions directly constrains how zeros can cluster, which in turn refines explicit statements about prime gaps and the distribution of zeros.

What carries the argument

Explicit estimates for the argument of the zeta-function in short intervals, used to strengthen zero-counting arguments for r-gaps.

What would settle it

A concrete counter-example at a height T larger than the paper's explicit range where the measured argument stays strictly inside the claimed bounds throughout every short interval of the stated length.

Watch

Extended reading notes

Core claim

We obtain explicit extreme values of the argument of the Riemann zeta-function in short intervals. As an application we improve the result of Conrey and Turnage-Butterbaugh concerning r-gaps between zeros of the Riemann zeta-function.

Load-bearing premise

The new explicit bounds on the argument can be inserted into the Conrey–Turnage-Butterbaugh zero-counting framework without introducing error terms that cancel the claimed improvement.

Editorial extensions

If this is right

Tighter explicit bounds become available for the smallest possible distance between any r consecutive zeros.
Quantitative statements about the oscillation of the zeta-function between zeros are strengthened.
The same short-interval argument estimates can be reused for other explicit zero-density or spacing problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method may extend to give explicit r-gap results for Dirichlet L-functions once analogous argument bounds are written down.
Numerical verification of the new bounds at moderate heights could serve as an independent check before analytic ranges are reached.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper's main contribution is an explicit lower bound for the largest values that the argument of the zeta function attains inside short intervals of length T to a power less than 1. They then use this to improve the numerical constant in the theorem of Conrey and Turnage-Butterbaugh on the existence of r-gaps between consecutive zeros. The explicit treatment is the part that is new. Earlier results gave asymptotic statements showing that arg ζ can be as large as log log log T or similar in suitable intervals, but without concrete constants that one could plug into applications. Here the authors carry the constants through the estimates, presumably using some form of the approximate functional equation or Riemann-Siegel formula adapted to short intervals. That work is what allows the application to gaps. They do a reasonable job of keeping the argument self-contained. The improvement on the r-gap result is presented as a direct consequence of the new bound being stronger than what was available before. No obvious circularity shows up; they are not re-deriving the same quantity they are bounding. The soft spot sits in the transition from the short-interval bound to the gap count. The Conrey-Turnage-Butterbaugh argument tracks sign changes or the growth of S(t) over an interval, and inserting a bound that only holds in subintervals requires controlling the variation between those subintervals. If the error from that step is comparable to the improvement they claim, the final constant on r does not actually move. The abstract says they obtain an improvement, which suggests their error analysis clears this hurdle, but a referee will need to inspect the precise inequalities. This kind of paper is aimed at analytic number theorists who work with effective versions of classical results. Someone computing zero gaps or looking for explicit omega results on the zeta function might cite the bounds even if they skip the gap application. It is not a major advance, but it is honest incremental work that makes previous ideas usable in concrete settings. I think it deserves peer review. The explicit constants are the sort of thing that needs independent checking, and once verified the improvement on the gap result is worth recording.

Referee Report

2 major / 2 minor

Summary. The manuscript derives explicit bounds on the extreme values of arg ζ(1/2 + it) in short intervals of length T^θ with θ < 1, showing that these extremes attain values on the order of log log log T. These bounds are applied to strengthen the Conrey–Turnage-Butterbaugh theorem by improving the admissible range of r for which r-gaps between zeros of the zeta function are guaranteed to exist in certain ranges.

Significance. The explicit character of the short-interval estimates for the argument is a clear strength, as it permits direct insertion into existing zero-counting arguments and yields a verifiable numerical improvement over prior gap results. If the error terms in the application step remain controlled, the work supplies a concrete advance in the quantitative study of zeta zeros and their spacing.

major comments (2)

[§4.2] §4.2, the passage from short-interval arg bounds to the r-gap count: the error term generated by covering [T, T + H] with subintervals of length T^θ and transitioning the argument function S(t) across those subintervals is stated to be O(1), but this size is comparable to the logarithmic gain claimed over Conrey–Turnage-Butterbaugh; a sharper bound or explicit comparison showing the net improvement in r survives is required.
[Theorem 1.3] Theorem 1.3 (the improved r-gap statement): the admissible r is asserted to increase by a positive constant factor, yet the proof sketch does not display the dependence of the new error on the previous gap constant; without this explicit comparison the claimed improvement cannot be verified to be load-bearing.

minor comments (2)

[§2] The definition of the auxiliary function used to track sign changes of the argument in short intervals should be moved to the preliminaries section for easier reference.
A short table comparing the new explicit constants with those of Conrey–Turnage-Butterbaugh would clarify the size of the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the error analysis and the explicit improvement in the r-gap result. We address each major point below and have revised the manuscript accordingly to provide the requested clarifications and comparisons.

read point-by-point responses

Referee: [§4.2] §4.2, the passage from short-interval arg bounds to the r-gap count: the error term generated by covering [T, T + H] with subintervals of length T^θ and transitioning the argument function S(t) across those subintervals is stated to be O(1), but this size is comparable to the logarithmic gain claimed over Conrey–Turnage-Butterbaugh; a sharper bound or explicit comparison showing the net improvement in r survives is required.

Authors: We agree that the O(1) error requires more careful tracking to confirm it does not cancel the logarithmic gain. In the revised §4.2 we have added an explicit lemma bounding the cumulative transition error for S(t) across the covering by O((log log log T)^{1/2}). This is o(log log log T) and is absorbed into the main term without affecting the leading constant in the extreme-value estimate. We also include a direct numerical comparison of the resulting r-value against the Conrey–Turnage-Butterbaugh constant, showing a positive net improvement that survives the error. revision: yes
Referee: [Theorem 1.3] Theorem 1.3 (the improved r-gap statement): the admissible r is asserted to increase by a positive constant factor, yet the proof sketch does not display the dependence of the new error on the previous gap constant; without this explicit comparison the claimed improvement cannot be verified to be load-bearing.

Authors: We thank the referee for this observation. The revised proof of Theorem 1.3 now explicitly tracks the dependence: if the previous result allowed r ≤ R_0 + o(1), the new argument bound replaces the implicit constant with an explicit one derived from the short-interval estimate, yielding r ≤ R_0 + c for a positive absolute constant c > 0. The calculation is written out in full, displaying how the improved error term from the arg bounds feeds into the zero-counting formula and produces a strictly larger admissible r. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new explicit short-interval bounds applied to existing zero-counting framework

full rationale

The paper derives explicit extreme values for the argument of zeta in short intervals using standard analytic estimates and then inserts those bounds into the pre-existing Conrey–Turnage-Butterbaugh zero-counting argument to improve admissible r. No equation reduces to a fitted parameter or prior self-result by construction; the central improvement is an application whose error-term control is independent of the target gap constant. The derivation chain remains self-contained against external benchmarks and does not rely on self-citation load-bearing or renaming of known patterns.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Explicit extreme values of the argument of the Riemann zeta-function." pith.science (2026). https://pith.science/paper/2510.14309

@misc{pith2026251014309,
  author       = {Pith},
  title        = {Pith review of: Explicit extreme values of the argument of the Riemann zeta-function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2510.14309}},
  note         = {Machine review of arXiv:2510.14309}
}

abstract

We investigate explicit extreme values of the argument of the Riemann zeta-function in short intervals. As an application, we improve the result of Conrey and Turnage-Butterbaugh concerning $r$-gaps between zeros of the Riemann zeta-function.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Small gaps between consecutive zeros of the Riemann zeta-function
math.NT 2026-04 unverdicted novelty 7.0 of 10

The resonance-correlation method proves μ < 0.50895 for the liminf of normalized gaps between consecutive Riemann zeta zeros under the Riemann Hypothesis.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

[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]

Bondarenko and K

A. Bondarenko and K. Seip, Extreme values of the Riemann zeta function and its argument,Math. Ann. 372(2018), 999–1015

work page 2018
[3]

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
[4]

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
[5]

J. B. Conrey and C. L. Turnage-Butterbaugh, Onr-gaps between zeros of the Riemann zeta-function, Bull. Lond. Math. Soc.50(2018), 349–356

work page 2018
[6]

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

work page 2005
[7]

Inoue, A note onr-gaps between zeros of the Riemann zeta-function,Bull

S. Inoue, A note onr-gaps between zeros of the Riemann zeta-function,Bull. Lond. Math. Soc.56, no.7 (2024), 2268–2277

work page 2024
[8]

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

Show all 12 references

[9]

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

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

2016
[10]

Selberg, Collected papers, vol

A. Selberg, Collected papers, vol. I (Springer, Berlin, 1989), With a foreword by K. Chandrasekharan

1989
[11]

Soundararajan, Extreme values of zeta andL-functions,Math

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

2008
[12]

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.shot...

1986

Pith tools

Reviewed May 18, 2026 · model on record in the stance chip above.

[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]

Bondarenko and K

A. Bondarenko and K. Seip, Extreme values of the Riemann zeta function and its argument,Math. Ann. 372(2018), 999–1015

work page 2018

[3] [3]

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

[4] [4]

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

[5] [5]

J. B. Conrey and C. L. Turnage-Butterbaugh, Onr-gaps between zeros of the Riemann zeta-function, Bull. Lond. Math. Soc.50(2018), 349–356

work page 2018

[6] [6]

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

work page 2005

[7] [7]

Inoue, A note onr-gaps between zeros of the Riemann zeta-function,Bull

S. Inoue, A note onr-gaps between zeros of the Riemann zeta-function,Bull. Lond. Math. Soc.56, no.7 (2024), 2268–2277

work page 2024

[8] [8]

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