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arxiv: 2604.25579 · v1 · submitted 2026-04-28 · 🧮 math.NT · math.PR

Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function

Louis-Pierre Arguin , Emma Bailey , Asher Roberts This is my paper

Pith reviewed 2026-05-07 15:02 UTC · model grok-4.3

classification 🧮 math.NT math.PR MSC 11M06
keywords Riemann zeta functionlarge deviationsmomentsRiemann hypothesiscritical lineupper bounds
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The pith

Assuming the Riemann Hypothesis, the proportion of times t in [T,2T] where |zeta(1/2 + it)| exceeds (log T)^k is at most C_k (log T)^{-k^2} / sqrt(log log T) for k>0.

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

The paper proves a conditional upper bound on the frequency of large values taken by the Riemann zeta function along the critical line. It shows that, under the Riemann Hypothesis, the set of t where |zeta(1/2 + it)| is larger than (log T)^k has measure bounded by C_k times (log T) to the power -k squared, divided by the square root of log log T, with C_k growing double-exponentially in k. This tail bound immediately yields an upper estimate for the 2k-moments of |zeta(1/2 + it)| of size C_k (log T)^{k^2}, matching a prior result of Harper. A reader would care because such control limits how extreme the fluctuations of zeta can be and thereby constrains the distribution of its zeros and values in the critical strip.

Core claim

Assuming the Riemann Hypothesis, we show that for k>0, (1/T) meas{t in [T,2T]: |zeta(1/2 + i t)| > (log T)^k} is at most C_k (log T)^{-k^2} / sqrt(log log T), where C_k = exp(e^{c k}) for an absolute constant c>0. This implies that the 2k-moments of |zeta| on the critical line are bounded above by C_k (log T)^{k^2}. The argument proceeds via a recursive scheme that iteratively reduces the large-deviation problem to moment estimates.

What carries the argument

A recursive scheme that reduces large-deviation estimates for |zeta(1/2 + it)| to successive moment bounds, obtained by combining the method of Soundararajan with the recursive framework developed by one author together with Bourgade and Radziwill.

If this is right

  • The 2k-moments of |zeta(1/2 + it)| for t in [T,2T] are at most C_k (log T)^{k^2}.
  • The same large-deviation bound recovers the moment upper bound previously proved by Harper.
  • The estimate applies for every fixed k>0 and yields an explicit double-exponential dependence of the constant C_k on k.

Where Pith is reading between the lines

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

  • The bound supplies a concrete tail decay rate that can be inserted into other estimates involving the maximum size of |zeta| on short intervals.
  • The recursive method may extend directly to analogous large-deviation questions for other families of L-functions whose moments are accessible by similar techniques.
  • If the constant C_k can be improved, the resulting moment bounds would become sharper for large but fixed k.

Load-bearing premise

The Riemann Hypothesis holds.

What would settle it

A numerical or analytic check for some fixed k and sufficiently large T that finds the actual measure of t in [T,2T] with |zeta(1/2 + it)| > (log T)^k to be strictly larger than C_k (log T)^{-k^2} / sqrt(log log T) while the Riemann Hypothesis is assumed.

read the original abstract

Assuming the Riemann Hypothesis, we show that for $k>0$ $$ \frac{1}{T}\text{meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}}, $$ where $C_k=\exp(e^{ck})$ for some absolute constant $c>0$. This implies that the $2k$-moments of $|\zeta|$ are bounded above by $C_k(\log T)^{k^2}$, recovering the bound of Harper. The proof relies on the recursive scheme of one of the authors with Bourgade and Radziwill (2020), and combines ideas of Soundararajan (2009) and Harper (2013).

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.

Referee Report

0 major / 2 minor

Summary. Assuming the Riemann Hypothesis, the paper establishes that for any fixed k > 0 the proportion of t in [T, 2T] for which |ζ(1/2 + it)| > (log T)^k satisfies 1/T meas{...} ≤ C_k (log T)^{-k²} / √(log log T) with C_k = exp(e^{c k}) for an absolute constant c > 0. The same tail bound is integrated to recover the upper bound ∫_T^{2T} |ζ(1/2 + it)|^{2k} dt ≪ C_k T (log T)^{k²} for the 2k-th moments, matching the order obtained by Harper.

Significance. If the derivation is correct, the manuscript supplies a new route to conditional large-deviation estimates for zeta by merging the 2020 Bourgade–Radziwiłł–Soundararajan recursive scheme with Soundararajan’s and Harper’s tail techniques. The resulting explicit (though rapidly growing) constant C_k and the clean passage from tail to moment via integration by parts constitute a modest but useful technical contribution to the study of the distribution of large values of zeta under RH.

minor comments (2)
  1. [Introduction] §1 (Introduction): the statement that the new bound 'recovers the bound of Harper' should be accompanied by a one-sentence comparison of the dependence of the implied constant on k, to make clear what is gained or lost relative to the 2013 argument.
  2. The recursive scheme is invoked at several points; a short paragraph summarizing the precise input taken from the 2020 paper (e.g., which moment or tail estimate is used as the base case) would improve readability for readers unfamiliar with that work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately reflects the main results and the significance section correctly identifies the technical contribution arising from combining the Bourgade–Radziwiłł–Soundararajan recursive scheme with tail estimates of Soundararajan and Harper. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation adds independent content under RH

full rationale

The paper assumes the Riemann Hypothesis and invokes the 2020 recursive scheme (Arguin-Bourgade-Radziwiłł) as an established tool, then combines it with independent ideas from Soundararajan (2009) and Harper (2013) to derive a new conditional large-deviation tail. The passage from tail bound to moment upper bound is a standard integration-by-parts argument on the distribution function. No equation or claim reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the central result has independent mathematical content beyond the cited prior work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Depends on Riemann Hypothesis as external assumption and on recursive scheme from prior self-cited work; only free parameter is absolute constant c in C_k; no new entities introduced.

free parameters (1)
  • absolute constant c
    Defines C_k = exp(e^{c k}); asserted to exist but no value or fitting given.
axioms (1)
  • domain assumption Riemann Hypothesis
    Explicitly assumed to derive the large-deviation bound.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

  1. [1]

    Arguin, P

    [ABR20] L.-P. Arguin, P. Bourgade, and M. Radziwi l l. The Fyodorov-Hiary-Keating con- jecture. I. Preprint arXiv: 2007.00988,

    work page arXiv 2007

  2. [2]

    Arguin, P

    [ABR23] L.-P. Arguin, P. Bourgade, and M. Radziwi l l. The Fyodorov-Hiary-Keating con- jecture. II. Preprint arXiv: arXiv:2307.00982,

    work page arXiv

  3. [3]

    Arguin and N

    [AC26] L.-P. Arguin and N. Creighton. Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line. Preprint arXiv:2603.01711,

    work page arXiv

  4. [4]

    Arguin and J

    [AH24] L.-P. Arguin and J. Hamdan. The Fyodorov-Hiary-Keating conjecture on meso- scopic intervals. Preprint arXiv:2405.06474,

    work page arXiv

  5. [5]

    [Har13] A. J. Harper. Sharp conditional bounds for moments of the Riemann zeta function. Preprint arXiv: 1305.4618,

    work page arXiv

  6. [6]

    [Har19] A. J. Harper. On the partition function of the Riemann zeta function, and the Fyodorov–Hiary–Keating conjecture. Preprint arXiv:1906.05783,

    work page arXiv 1906

  7. [7]

    Radziwi l l

    [Rad11] M. Radziwi l l. Large Deviations in Selberg’s Central Limit Theorem. Preprint arxiv.org/abs/1108.5092,

    work page arXiv

  8. [8]

    EMS Press, Berlin, [2023]©2023

    Plenary lectures, pages 1260–1310. EMS Press, Berlin, [2023]©2023. [Tao24] T. Tao. Optimization of the implicit constant for the upper bounds for moments of the Riemann zeta function. Preprint arXiv: arXiv:2407.20023,

    work page arXiv 2023