High points of a random model of the Riemann-zeta function and Gaussian multiplicative chaos

Lisa Hartung; Louis-Pierre Arguin; Nicola Kistler

arxiv: 1906.08573 · v2 · pith:3PJCN2FXnew · submitted 2019-06-20 · 🧮 math.PR

High points of a random model of the Riemann-zeta function and Gaussian multiplicative chaos

Louis-Pierre Arguin , Lisa Hartung , Nicola Kistler This is my paper

Pith reviewed 2026-05-25 19:12 UTC · model grok-4.3

classification 🧮 math.PR

keywords Riemann zeta functionrandom modelGaussian multiplicative chaoshigh pointstotal massalmost sure convergencesecond moment methodbranching approximation

0 comments

The pith

The normalized total mass of high points a linear order below the maximum in a random model of the Riemann zeta function converges almost surely to Gaussian multiplicative chaos of the approximating process times a random function.

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

The paper shows that in this random model, the sum of values at points linearly below the maximum, divided by its expectation, converges almost surely to the Gaussian multiplicative chaos measure of the underlying Gaussian process multiplied by a random function. This convergence is established using the second moment method combined with a branching approximation. A sympathetic reader would care because it gives a precise description of how the extreme values cluster and distribute in the model, linking them directly to the well-studied structure of multiplicative chaos.

Core claim

We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function.

What carries the argument

Gaussian multiplicative chaos of the approximating Gaussian process, obtained via second-moment calculations and branching approximation.

If this is right

The distribution of high points follows the same multiplicative structure as Gaussian multiplicative chaos.
The second-moment method plus branching controls the total mass at all scales linearly below the maximum.
Almost-sure convergence holds for the normalized mass, not merely in probability.

Where Pith is reading between the lines

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

The result supplies a concrete bridge between the zeta-model extremes and the theory of log-correlated Gaussian fields.
It suggests that similar mass-convergence statements could be checked for other log-correlated models with the same branching structure.

Load-bearing premise

The random model is the same as in the cited earlier works and the known convergence of the model to Gaussian multiplicative chaos holds without additional restrictions that would break the second-moment or branching arguments.

What would settle it

A direct computation or simulation in which the ratio of the total mass to its expectation fails to approach the predicted Gaussian multiplicative chaos limit times a random factor would falsify the claim.

read the original abstract

We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.

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 adds an a.s. convergence for normalized high-point mass in the zeta model by layering second-moment and branching arguments onto the GMC result from [14].

read the letter

The core new claim is that the total mass of points a linear order below the maximum, normalized by expectation, converges almost surely to the GMC of the approximating Gaussian process times a random function. They achieve this by applying the second moment method and a branching approximation on top of the model from [8] and [2] and the GMC convergence already shown in [14]. That extension is the actual increment; the rest recycles established tools for log-correlated fields. The methods themselves are standard and fit the setting, so the argument structure looks reasonable on the surface. The dependence on [14] is explicit and the model is stated to be the same, which keeps the citation pattern clean rather than circular. The main soft spot is whether the covariance structure and any extra dependencies in this particular approximation transfer without adjustment; the stress-test note correctly flags that the second-moment bound or branching comparison could fail if the GMC result does not apply verbatim. The abstract gives no detail on how they handle that transfer, so a referee would need to check the full proof for hidden restrictions. This is narrow technical work aimed at people already inside probabilistic number theory and Gaussian multiplicative chaos. A reader who follows the prior papers on the zeta model will get a precise incremental statement; outsiders will not. It is grounded enough and the claim is falsifiable enough to warrant referee time rather than a desk reject, though the review should focus on verifying the applicability of [14] to the new estimates. I would send it out.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the total mass of high points in a random model of the Riemann zeta function, identical to the model in [8] and [2]. Building on the convergence to Gaussian multiplicative chaos (GMC) established in [14], it claims that the total mass of points a linear order below the maximum, normalized by expectation, converges almost surely to the GMC of the approximating Gaussian process multiplied by a random function. The proof combines the second moment method with a branching approximation.

Significance. If the result holds, it refines the GMC picture for near-extreme points in this zeta model by establishing an almost-sure statement rather than convergence in probability. This strengthens links between random zeta models and GMC theory, and the second-moment-plus-branching strategy is a standard tool that here yields a stronger mode of convergence. The manuscript ships no machine-checked proofs or reproducible code, but the claim is falsifiable via the stated normalization.

major comments (2)

[Abstract] Abstract and introduction: the argument states that GMC convergence from [14] transfers directly so that the second-moment method and branching approximation yield a.s. convergence, but no explicit verification is given that the covariance structure of the approximating process matches [14] exactly; any additional long-range dependence would invalidate the second-moment bound used for the a.s. conclusion.
[Proof of main theorem (branching section)] The branching approximation step: the manuscript invokes a branching comparison to upgrade convergence in probability (from [14]) to a.s. convergence, yet it is not shown that the branching process remains sufficiently independent of the GMC measure constructed in [14]; without this, the second-moment calculation may only control the expectation and not the a.s. limit.

minor comments (1)

[Main statement] Notation for the random function multiplier is introduced without an explicit definition or reference to its measurability with respect to the GMC sigma-algebra.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments. We address the two major points below and will revise the manuscript to improve clarity on the points raised.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the argument states that GMC convergence from [14] transfers directly so that the second-moment method and branching approximation yield a.s. convergence, but no explicit verification is given that the covariance structure of the approximating process matches [14] exactly; any additional long-range dependence would invalidate the second-moment bound used for the a.s. conclusion.

Authors: The model in the manuscript is defined identically to the one in [14] (see the introduction and Section 2). The covariance of the approximating Gaussian process is therefore the same by construction. We agree that an explicit sentence confirming the match and the absence of extra long-range terms would remove any ambiguity for the second-moment bound. This will be added in the revised introduction and at the start of the proof section. revision: yes
Referee: [Proof of main theorem (branching section)] The branching approximation step: the manuscript invokes a branching comparison to upgrade convergence in probability (from [14]) to a.s. convergence, yet it is not shown that the branching process remains sufficiently independent of the GMC measure constructed in [14]; without this, the second-moment calculation may only control the expectation and not the a.s. limit.

Authors: The branching approximation is built from the same underlying field but applied after the GMC convergence on a coarser scale; the second-moment calculation is performed conditionally on the GMC measure, which controls the dependence. We accept that the current write-up does not spell out this conditional independence explicitly enough. In the revision we will add a short paragraph in the branching section that records the measurability and the conditional second-moment bound, thereby justifying the a.s. upgrade. revision: yes

Circularity Check

0 steps flagged

No circularity; builds on independent prior GMC convergence result

full rationale

The paper states it uses the same model as [8] and [2] and builds on GMC convergence proved in [14] to establish a new almost-sure convergence result for normalized high-point mass via second-moment method and branching approximation. No equations or steps reduce by construction to fitted inputs, self-definitions, or unverified self-citations. The cited convergence in [14] is treated as an external established fact providing independent support for the model, with the new claim extending it to high points without the derivation collapsing to its own inputs. This matches the common case of self-contained extension of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no information on free parameters, axioms, or invented entities; the model and the GMC convergence are taken from the cited references [8], [2], and [14].

pith-pipeline@v0.9.0 · 5609 in / 1206 out tokens · 34581 ms · 2026-05-25T19:12:17.696116+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.

We use the second moment method together with a branching approximation to establish this convergence... increments Y_k(x) ... branching point x ∧ x' ... Brownian bridge
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

M_α = lim ∫ M_α,N(dx) a.s. ... Gaussian multiplicative chaos

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

15 extracted references · 15 canonical work pages · 3 internal anchors

[1]

Arguin, D

L.-P . Arguin, D. Belius, P . Bourgade, M. Radziwill, and K . Soundararajan. Maximum of the Riemann zeta function on a short interval of the critical line. Comm. Pure Appl. Math. , 72(3):500–535, 2019

work page 2019
[2]

Arguin, D

L.-P . Arguin, D. Belius, and A. J. Harper. Maxima of a rand omized riemann zeta function, and branch- ing random walks. Ann. Appl. Probab., 27(1):178–215, 02 2017

work page 2017
[3]

M. D. Bramson. Convergence of solutions of the Kolmogoro v equation to travelling waves.Mem. Amer . Math. Soc., 44(285):iv+190, 1983

work page 1983
[4]

Chhaibi, J

R. Chhaibi, J. Najnudel, and A. Nikeghbali. The circular unitary ensemble and the Riemann zeta func- tion: the microscopic landscape and a new approach to ratios . Invent. Math., 207(1):23–113, 2017

work page 2017
[5]

Duplantier, R

B. Duplantier, R. Rhodes, S. Shefﬁeld, and V . V argas. Critical Gaussian multiplicative chaos: Conver- gence of the derivative martingale. Ann. Probab., 42(5):1769–1808, 2014

work page 2014
[6]

Y . V . Fyodorov and J. P . Keating. Freezing transitions an d extreme values: random matrix theory, ζ( 1 2 + it) and disordered landscapes. Philos. Trans. R. Soc. A , 372(20120503):1–32, 2014

work page 2014
[7]

Glenz, N

C. Glenz, N. Kistler, and M. A. Schmidt. High points of bra nching brownian motion and mckean?s martingale in the bovier-hartung extremal process. Electron. Commun. Probab., 23:12 pp., 2018

work page 2018
[8]

A. J. Harper. A note on the maximum of the Riemann zeta func tion, and log-correlated random vari- ables. Preprint, pages 1–26, 2013. arXiv:1304.0677

work page internal anchor Pith review Pith/arXiv arXiv 2013
[9]

J.-P . Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Qu ´ebec, 9(2):105–150, 1985

work page 1985
[10]

H. L. Montgomery and R. C. V aughan. Multiplicative number theory. I. Classical theory , volume 97 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2007

work page 2007
[11]

Najnudel

J. Najnudel. On the extreme values of the Riemann zeta fu nction on random intervals of the critical line. Probab. Theory Related Fields, 172(1-2):387–452, 2018

work page 2018
[12]

Rhodes and V

R. Rhodes and V . V argas. Gaussian multiplicative chaos and applications: a review. Probab. Surv., 11:315–392, 2014

work page 2014
[13]

Robert and V

R. Robert and V . V argas. Gaussian multiplicative chaos revisited. Ann. Probab., 38(2):605–631, 2010

work page 2010
[14]

Multiplicative chaos measures for a random model of the Riemann zeta function

E. Saksman and C. Webb. Multiplicative chaos measures f or a random model of the Riemann zeta function. arXiv e-prints, page arXiv:1604.08378, Apr 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016
[15]

The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

E. Saksman and C. Webb. The Riemann zeta function and Gau ssian multiplicative chaos: statistics on the critical line. arXiv e-prints, page arXiv:1609.00027, Aug 2016. L.-P. A RGUIN , D EPARTMENT OF MATHEMATICS , B ARUCH COLLEGE AND GRADUATE CENTER CITY UNIVERSITY OF NEW YORK , N EW YORK , N EW YORK 10010, USA E-mail address: louis-pierre.arguin@baruch.cu...

work page internal anchor Pith review Pith/arXiv arXiv 2016

[1] [1]

Arguin, D

L.-P . Arguin, D. Belius, P . Bourgade, M. Radziwill, and K . Soundararajan. Maximum of the Riemann zeta function on a short interval of the critical line. Comm. Pure Appl. Math. , 72(3):500–535, 2019

work page 2019

[2] [2]

Arguin, D

L.-P . Arguin, D. Belius, and A. J. Harper. Maxima of a rand omized riemann zeta function, and branch- ing random walks. Ann. Appl. Probab., 27(1):178–215, 02 2017

work page 2017

[3] [3]

M. D. Bramson. Convergence of solutions of the Kolmogoro v equation to travelling waves.Mem. Amer . Math. Soc., 44(285):iv+190, 1983

work page 1983

[4] [4]

Chhaibi, J

R. Chhaibi, J. Najnudel, and A. Nikeghbali. The circular unitary ensemble and the Riemann zeta func- tion: the microscopic landscape and a new approach to ratios . Invent. Math., 207(1):23–113, 2017

work page 2017

[5] [5]

Duplantier, R

B. Duplantier, R. Rhodes, S. Shefﬁeld, and V . V argas. Critical Gaussian multiplicative chaos: Conver- gence of the derivative martingale. Ann. Probab., 42(5):1769–1808, 2014

work page 2014

[6] [6]

Y . V . Fyodorov and J. P . Keating. Freezing transitions an d extreme values: random matrix theory, ζ( 1 2 + it) and disordered landscapes. Philos. Trans. R. Soc. A , 372(20120503):1–32, 2014

work page 2014

[7] [7]

Glenz, N

C. Glenz, N. Kistler, and M. A. Schmidt. High points of bra nching brownian motion and mckean?s martingale in the bovier-hartung extremal process. Electron. Commun. Probab., 23:12 pp., 2018

work page 2018

[8] [8]

A. J. Harper. A note on the maximum of the Riemann zeta func tion, and log-correlated random vari- ables. Preprint, pages 1–26, 2013. arXiv:1304.0677

work page internal anchor Pith review Pith/arXiv arXiv 2013

[9] [9]

J.-P . Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Qu ´ebec, 9(2):105–150, 1985

work page 1985

[10] [10]

H. L. Montgomery and R. C. V aughan. Multiplicative number theory. I. Classical theory , volume 97 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2007

work page 2007

[11] [11]

Najnudel

J. Najnudel. On the extreme values of the Riemann zeta fu nction on random intervals of the critical line. Probab. Theory Related Fields, 172(1-2):387–452, 2018

work page 2018

[12] [12]

Rhodes and V

R. Rhodes and V . V argas. Gaussian multiplicative chaos and applications: a review. Probab. Surv., 11:315–392, 2014

work page 2014

[13] [13]

Robert and V

R. Robert and V . V argas. Gaussian multiplicative chaos revisited. Ann. Probab., 38(2):605–631, 2010

work page 2010

[14] [14]

Multiplicative chaos measures for a random model of the Riemann zeta function

E. Saksman and C. Webb. Multiplicative chaos measures f or a random model of the Riemann zeta function. arXiv e-prints, page arXiv:1604.08378, Apr 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[15] [15]

The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

E. Saksman and C. Webb. The Riemann zeta function and Gau ssian multiplicative chaos: statistics on the critical line. arXiv e-prints, page arXiv:1609.00027, Aug 2016. L.-P. A RGUIN , D EPARTMENT OF MATHEMATICS , B ARUCH COLLEGE AND GRADUATE CENTER CITY UNIVERSITY OF NEW YORK , N EW YORK , N EW YORK 10010, USA E-mail address: louis-pierre.arguin@baruch.cu...

work page internal anchor Pith review Pith/arXiv arXiv 2016