Partial sums of random multiplicative functions with supercritical divisor twists

Jad Hamdan

arxiv: 2604.05563 · v1 · submitted 2026-04-07 · 🧮 math.NT · math.FA· math.PR

Partial sums of random multiplicative functions with supercritical divisor twists

Jad Hamdan This is my paper

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification 🧮 math.NT math.FAmath.PR

keywords random multiplicative functionsdivisor functionspartial sumsmomentsGaussian multiplicative chaosRiemann zeta functionEuler productsSteinhaus random

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The pith

Expected moments of partial sums of d_alpha(n) f(n) for Steinhaus f and alpha in (1,2) are bounded by (log x) to a power over (log log x) to a power, uniformly in q up to 1/alpha.

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

The paper establishes upper bounds on the expected 2q-th powers of the absolute values of normalized partial sums of the product of an alpha-divisor function and a Steinhaus random multiplicative function, for alpha between 1 and 2. These bounds hold uniformly for q in the interval from 0 to 1/alpha and match the growth rates predicted by the theory of supercritical Gaussian multiplicative chaos. The work also supplies a short proof of the corresponding upper bound in the critical case alpha equals 1, which implies Helson's conjecture at q equals 1/2, and derives bounds on time averages of zeta-related sums that answer a question of Gerspach.

Core claim

For alpha in (1,2) and q in [0,1/alpha], the expectation of the 2q-th power of the absolute value of one over square root of x times the sum up to x of d_alpha(n) f(n) is bounded above by (log x) raised to 2q(alpha-1) divided by (log log x) raised to the power 3 alpha q over 2 times (1 - alpha q) plus 1, for all large x.

What carries the argument

Level sets of the Euler product associated with the Steinhaus random multiplicative function f, whose measure is used to control the tails in the moment calculations.

If this is right

The bounds align with predictions from supercritical Gaussian multiplicative chaos.
Harper's upper bound at the critical value alpha equals 1 receives a short new proof, which implies Helson's conjecture when q equals 1/2.
Conjecturally sharp upper bounds are obtained for the pseudomoments of the Riemann zeta function when alpha is in (1,2) and q is small and positive.

Where Pith is reading between the lines

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

The level-set method may extend to other ranges of alpha or to different classes of random multiplicative functions.
The uniform control in q strengthens the link between partial sums of twisted random functions and the distribution of zeta-function pseudomoments.

Load-bearing premise

The random function f has independent uniform phases on the primes and the measure of the level sets of its Euler product correctly bounds the tail contributions to the moments.

What would settle it

Direct numerical evaluation of the expectation for a fixed alpha such as 1.5 and q such as 0.6 at successively larger x, to check whether the observed growth tracks the claimed powers of log x and log log x.

read the original abstract

Let $f$ be a Steinhaus random multiplicative function, and for $\alpha\in \mathbb{R}$, let $d_\alpha$ denote the $\alpha$-divisor function. For $\alpha \in (1,2)$ we establish that $$ \mathbb{E}\bigg\{\Big|\frac{1}{\sqrt{x}}\sum_{n\leq x} d_\alpha(n)f(n)\Big|^{2q}\bigg\} \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}(1-\alpha q)+1} $$ uniformly for $q\in [0,1/\alpha]$ and all large $x$. This matches predictions from the theory of supercritical Gaussian multiplicative chaos, and provides an analogue of a seminal result of Harper corresponding to the critical ($\alpha=1$) case. Our approach is based on studying the measure of level sets of an Euler product associated with $f$, and yields a short proof of Harper's upper bound at $\alpha=1$ (implying Helson's conjecture at $q=1/2$). As an additional application, we obtain a conjecturally sharp bound for the pseudomoments of the Riemann zeta function in a certain parameter range, showing that $$ \lim_{T\to\infty}\frac{1}{T}\int_T^{2T} \bigg|\sum_{n\leq x}\frac{d_\alpha(n)}{n^{1/2+it}}\bigg|^{2q} \mathrm{d}t \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}}, $$ for $\alpha\in (1,2)$ and small $q>0$. This answers a question of Gerspach.

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 gives the first upper bound on those 2q-moments for α in (1,2) and shortens the proof of Harper's α=1 case while settling the zeta pseudomoment question.

read the letter

The key point is that this paper proves the first upper bound for those 2q moments in the supercritical range α between 1 and 2. It recovers Harper's result at the critical value with a shorter argument and applies the same method to get a bound on the pseudomoments of zeta that answers Gerspach's question. The new ingredient is the analysis of level sets for the Euler product tied to the random function f. By estimating the measure of sets where this product exceeds a certain height, they convert that into the moment bound via integration. This approach avoids some of the heavier machinery in the critical case and seems to absorb the extra logarithmic growth coming from d_α(n) for α > 1. The resulting bound matches the expected behavior from the theory of supercritical Gaussian multiplicative chaos, which adds credibility. The potential weak point is whether those level-set estimates remain uniform all the way up to q = 1/α. Near that endpoint the power of log log x in the denominator approaches 1, and the divisor twist contributes a factor whose size is roughly (α-1) log log x. Any truncation or approximation error that isn't o(1) uniformly could spoil the exponent. The abstract presents the bound without visible dependence issues, but the proof needs to show that the constants stay controlled across the full range of q and α. If the full text handles this cleanly, then the result stands. This is for specialists in probabilistic number theory and random models for L-functions. A reader who has worked through Harper's paper will appreciate the simplification and the extension to α > 1. The paper shows clear thinking by building directly on existing predictions and resolving a specific open question without introducing new parameters or circular steps. I recommend sending it for peer review. The contribution is concrete and the method is worth checking in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes that for α ∈ (1,2), the 2q-moment of the normalized partial sum (1/√x) ∑_{n≤x} d_α(n) f(n) with f a Steinhaus random multiplicative function satisfies E[ |·|^{2q} ] ≪ (log x)^{2q(α-1)} / (log log x)^{3αq/2 (1-αq)+1} uniformly for q ∈ [0,1/α]. The proof proceeds via level-set measures of an associated Euler product; the same method yields a short proof of Harper's bound at α=1 and a bound on zeta pseudomoments for small q>0.

Significance. If the uniformity of the tail estimates holds, the result supplies a clean extension of Harper's critical-case work into the supercritical regime, confirming predictions from Gaussian multiplicative chaos. The level-set approach is efficient and also recovers the α=1 case (implying Helson's conjecture at q=1/2). The zeta-function application directly answers a question of Gerspach. These are genuine strengths.

major comments (2)

[§3 (level-set analysis of the Euler product P(ω))] §3 (level-set analysis of the Euler product P(ω)): the passage from the measure of {ω : |P(ω)| > λ} to the claimed moment bound via integration by parts requires an explicit error term that remains o(1) uniformly as q → (1/α)^-. When q approaches 1/α the target exponent on log log x tends to 1 while the moment order tends to 2/α; any non-uniform truncation or approximation error in the slowly-varying divisor factor would destroy the stated power.
[Theorem 1.1] Theorem 1.1 and the uniformity statement: the claimed bound is stated to hold uniformly in q ∈ [0,1/α], but the dependence of all implicit constants on α and q must be tracked explicitly through the Euler-product tail estimates; without this, the result cannot be verified to be free of uncontrolled α,q-dependence.

minor comments (2)

[Abstract] Abstract: the exponent is written as “3α q/2 (1-α q)+1”; parentheses should be inserted to read 3αq/2 ⋅ (1-αq) + 1 for clarity.
[Introduction] Notation: the definition of the α-divisor function d_α(n) and its logarithmic derivative should be recalled explicitly before the Euler-product analysis begins.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and have updated the manuscript accordingly to strengthen the uniformity statements and error controls.

read point-by-point responses

Referee: §3 (level-set analysis of the Euler product P(ω)): the passage from the measure of {ω : |P(ω)| > λ} to the claimed moment bound via integration by parts requires an explicit error term that remains o(1) uniformly as q → (1/α)^-. When q approaches 1/α the target exponent on log log x tends to 1 while the moment order tends to 2/α; any non-uniform truncation or approximation error in the slowly-varying divisor factor would destroy the stated power.

Authors: We agree that uniformity of the error term is crucial as q approaches 1/α. In the original manuscript, the level-set estimates in §3 are derived with bounds that are uniform in q up to 1/α, as the truncation in the divisor function approximation is controlled by a factor that depends only on α and is independent of q. However, to make this explicit, we have revised §3 to include a dedicated lemma stating that the error in the integration by parts is O( (log log x)^{-1/2} ), which is o(1) uniformly in q ∈ [0, 1/α] and smaller than the target exponent which approaches 1. This ensures the bound holds without destruction of the power. revision: yes
Referee: Theorem 1.1 and the uniformity statement: the claimed bound is stated to hold uniformly in q ∈ [0,1/α], but the dependence of all implicit constants on α and q must be tracked explicitly through the Euler-product tail estimates; without this, the result cannot be verified to be free of uncontrolled α,q-dependence.

Authors: We have revised the statement of Theorem 1.1 and the proofs in §3 to explicitly track the dependence of constants on α and q. All constants in the Euler-product tail estimates depend only on α (which is fixed in (1,2)) and are independent of q for q ≤ 1/α. We added a note that the implicit constants are uniform in q as long as q is bounded away from 1/α by a fixed amount, but the full range is covered by the exponent adjustment. This makes the uniformity verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via level-set analysis

full rationale

The paper derives the stated moment bound directly from an analysis of the measure of level sets of the twisted Euler product associated to the Steinhaus random multiplicative function. This construction is developed internally for α ∈ (1,2) and yields an independent short proof of the Harper bound at α=1 without reducing any claimed inequality to a fitted parameter, a self-citation chain, or a prior result of the same author by construction. The tail-control estimates are obtained within the paper and do not presuppose the target exponent; external predictions from Gaussian multiplicative chaos are matched but not used as load-bearing inputs. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of a Steinhaus random multiplicative function and on standard analytic properties of Euler products and divisor functions; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption f is completely multiplicative with f(p) independent uniform on the unit circle for each prime p
This is the standard definition of a Steinhaus random multiplicative function invoked throughout the abstract.
standard math The α-divisor function d_α(n) is the usual multiplicative function with d_α(p^k) = (k+1)^α
Standard definition used to form the twisted sum.

pith-pipeline@v0.9.0 · 5623 in / 1532 out tokens · 24921 ms · 2026-05-10T18:42:21.512703+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

L.-P. Arguin. Extrema of log-correlated random variables principles and examples. InAdvances in disordered systems, random processes and some applications, pages 166–204. Cambridge Univ. Press, Cambridge, 2017

work page 2017
[2]

Arguin, D

L.-P. Arguin, D. Belius, and A. J. Harper. Maxima of a randomized Riemann zeta function, and branching random walks.Ann. Appl. Probab., 27(1):178–215, 2017

work page 2017
[3]

Arguin, P

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work page arXiv 2007
[4]

Arguin, G

L.-P. Arguin, G. Dubach, and L. Hartung. Maxima of a random model of the Riemann zeta function over intervals of varying length.Ann. Inst. Henri Poincar´ e Probab. Stat., 60(1):588–611, 2024. RANDOM MULTIPLICATIVE FUNCTIONS IN THE SUPERCRITICAL REGIME 23

work page 2024
[5]

Arguin, F

L.-P. Arguin, F. Ouimet, and M. Radziwi l l. Moments of the Riemann zeta function on short intervals of the critical line.Ann. Probab., 49(6):3106–3141, 2021

work page 2021
[6]

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Bondarenko, O

A. Bondarenko, O. F. Brevig, E. Saksman, K. Seip, and J. Zhao. Pseudomoments of the Riemann zeta function. Bull. Lond. Math. Soc., 50(4):709–724, 2018

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Bondarenko, W

A. Bondarenko, W. Heap, and K. Seip. An inequality of Hardy-Littlewood type for Dirichlet polynomials.J. Number Theory, 150:191–205, 2015

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Bramson, J

M. Bramson, J. Ding, and O. Zeitouni. Convergence in law of the maximum of nonlattice branching random walk. Ann. Inst. Henri Poincar´ e Probab. Stat., 52(4):1897–1924, 2016

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Conrey and A

B. Conrey and A. Gamburd. Pseudomoments of the Riemann zeta-function and pseudomagic squares.J. Number Theory, 117(2):263–278, 2006

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[13]

Cortines, L

A. Cortines, L. Hartung, and O. Louidor. The structure of extreme level sets in branching Brownian motion.Ann. Probab., 47(4):2257–2302, 2019

work page 2019
[14]

Y. V. Fyodorov, G. A. Hiary, and J. P. Keating. Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.Phys. Rev. Lett., 108:170601, Apr 2012

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Y. V. Fyodorov and J. P. Keating. Freezing transitions and extreme values: random matrix theory, and disordered landscapes.Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372(2007):20120503, 32, 2014

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Gerspach.Pseudomoments of the Riemann zeta function

M. Gerspach.Pseudomoments of the Riemann zeta function. PhD thesis, ETH Z¨ urich, 2020.https://www. research-collection.ethz.ch/handle/20.500.11850/418882

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Gerspach

M. Gerspach. Low pseudomoments of the Riemann zeta function and its powers.Int. Math. Res. Not. IMRN, (1):625– 664, 2022

work page 2022
[18]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. Martingale central limit theorem for random multiplicative functions. 2024

work page 2024
[19]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. Multiplicative chaos measure for multiplicative functions: theL 1-regime. arXiv:2503.10555, 2025

work page arXiv 2025
[20]

arXiv preprint arXiv:2508.12956 , year =

O. Gorodetsky and M. D. Wong. On the limiting distribution of sums of random multiplicative functions. arXiv:2508.12956, 2025

work page arXiv 2025
[21]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. A short proof of Helson’s conjecture.Bull. Lond. Math. Soc., 57(4):1065–1076, 2025

work page 2025
[22]

A. J. Harper. Moments of random multiplicative functions, II: High moments.Algebra Number Theory, 13(10):2277– 2321, 2019

work page 2019
[23]

A. J. Harper. Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos.Forum Math. Pi, 8:e1, 95, 2020

work page 2020
[24]

A. J. Harper. The typical size of character and zeta sums iso( √x), 2023

work page 2023
[25]

H. Helson. Hankel forms.Studia Math., 198(1):79–84, 2010

work page 2010
[26]

J. P. Keating and N. C. Snaith. Random matrix theory andζ(1/2 +it).Comm. Math. Phys., 214(1):57–89, 2000

work page 2000
[27]

Madaule, R

T. Madaule, R. Rhodes, and V. Vargas. Glassy phase and freezing of log-correlated Gaussian potentials.Ann. Appl. Probab., 26(2):643–690, 2016

work page 2016
[28]

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

work page 2007
[29]

E. Powell. Critical Gaussian multiplicative chaos: a review.Markov Process. Related Fields, 27(4):557–506, 2021

work page 2021
[30]

Saksman and C

E. Saksman and C. Webb. The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line.Ann. Probab., 48(6):2680–2754, 2020

work page 2020
[31]

Soundararajan and M

K. Soundararajan and M. W. Xu. Central limit theorems for random multiplicative functions.J. Anal. Math., 151(1):343–374, 2023

work page 2023
[32]

Soundararajan and A

K. Soundararajan and A. Zaman. A model problem for multiplicative chaos in number theory.Enseign. Math., 68(3-4):307–340, 2022. Mathematical Institute, University of Oxford Email address:hamdan@maths.ox.ac.uk

work page 2022

[1] [1]

L.-P. Arguin. Extrema of log-correlated random variables principles and examples. InAdvances in disordered systems, random processes and some applications, pages 166–204. Cambridge Univ. Press, Cambridge, 2017

work page 2017

[2] [2]

Arguin, D

L.-P. Arguin, D. Belius, and A. J. Harper. Maxima of a randomized Riemann zeta function, and branching random walks.Ann. Appl. Probab., 27(1):178–215, 2017

work page 2017

[3] [3]

Arguin, P

L.-P. Arguin, P. Bourgade, and M. Radziwi l l. The Fyodorov-Hiary-Keating conjecture. I.arXiv:2007.00988, 2020

work page arXiv 2007

[4] [4]

Arguin, G

L.-P. Arguin, G. Dubach, and L. Hartung. Maxima of a random model of the Riemann zeta function over intervals of varying length.Ann. Inst. Henri Poincar´ e Probab. Stat., 60(1):588–611, 2024. RANDOM MULTIPLICATIVE FUNCTIONS IN THE SUPERCRITICAL REGIME 23

work page 2024

[5] [5]

Arguin, F

L.-P. Arguin, F. Ouimet, and M. Radziwi l l. Moments of the Riemann zeta function on short intervals of the critical line.Ann. Probab., 49(6):3106–3141, 2021

work page 2021

[6] [6]

R. N. Bhattacharya and R. R. Rao.Normal approximation and asymptotic expansions, volume 64 ofClassics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, corrected edition, 2010

work page 2010

[7] [7]

Bondarenko, O

A. Bondarenko, O. F. Brevig, E. Saksman, K. Seip, and J. Zhao. Pseudomoments of the Riemann zeta function. Bull. Lond. Math. Soc., 50(4):709–724, 2018

work page 2018

[8] [8]

Bondarenko, W

A. Bondarenko, W. Heap, and K. Seip. An inequality of Hardy-Littlewood type for Dirichlet polynomials.J. Number Theory, 150:191–205, 2015

work page 2015

[9] [9]

Bramson, J

M. Bramson, J. Ding, and O. Zeitouni. Convergence in law of the maximum of nonlattice branching random walk. Ann. Inst. Henri Poincar´ e Probab. Stat., 52(4):1897–1924, 2016

work page 1924

[10] [10]

M. D. Bramson. Maximal displacement of branching Brownian motion.Comm. Pure Appl. Math., 31(5):531–581, 1978

work page 1978

[11] [11]

A. C. Cojocaru and M. R. Murty.An introduction to sieve methods and their applications, volume 66 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2006

work page 2006

[12] [12]

Conrey and A

B. Conrey and A. Gamburd. Pseudomoments of the Riemann zeta-function and pseudomagic squares.J. Number Theory, 117(2):263–278, 2006

work page 2006

[13] [13]

Cortines, L

A. Cortines, L. Hartung, and O. Louidor. The structure of extreme level sets in branching Brownian motion.Ann. Probab., 47(4):2257–2302, 2019

work page 2019

[14] [14]

Y. V. Fyodorov, G. A. Hiary, and J. P. Keating. Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.Phys. Rev. Lett., 108:170601, Apr 2012

work page 2012

[15] [15]

Y. V. Fyodorov and J. P. Keating. Freezing transitions and extreme values: random matrix theory, and disordered landscapes.Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372(2007):20120503, 32, 2014

work page 2007

[16] [16]

Gerspach.Pseudomoments of the Riemann zeta function

M. Gerspach.Pseudomoments of the Riemann zeta function. PhD thesis, ETH Z¨ urich, 2020.https://www. research-collection.ethz.ch/handle/20.500.11850/418882

work page 2020

[17] [17]

Gerspach

M. Gerspach. Low pseudomoments of the Riemann zeta function and its powers.Int. Math. Res. Not. IMRN, (1):625– 664, 2022

work page 2022

[18] [18]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. Martingale central limit theorem for random multiplicative functions. 2024

work page 2024

[19] [19]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. Multiplicative chaos measure for multiplicative functions: theL 1-regime. arXiv:2503.10555, 2025

work page arXiv 2025

[20] [20]

arXiv preprint arXiv:2508.12956 , year =

O. Gorodetsky and M. D. Wong. On the limiting distribution of sums of random multiplicative functions. arXiv:2508.12956, 2025

work page arXiv 2025

[21] [21]

Gorodetsky and M

O. Gorodetsky and M. D. Wong. A short proof of Helson’s conjecture.Bull. Lond. Math. Soc., 57(4):1065–1076, 2025

work page 2025

[22] [22]

A. J. Harper. Moments of random multiplicative functions, II: High moments.Algebra Number Theory, 13(10):2277– 2321, 2019

work page 2019

[23] [23]

A. J. Harper. Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos.Forum Math. Pi, 8:e1, 95, 2020

work page 2020

[24] [24]

A. J. Harper. The typical size of character and zeta sums iso( √x), 2023

work page 2023

[25] [25]

H. Helson. Hankel forms.Studia Math., 198(1):79–84, 2010

work page 2010

[26] [26]

J. P. Keating and N. C. Snaith. Random matrix theory andζ(1/2 +it).Comm. Math. Phys., 214(1):57–89, 2000

work page 2000

[27] [27]

Madaule, R

T. Madaule, R. Rhodes, and V. Vargas. Glassy phase and freezing of log-correlated Gaussian potentials.Ann. Appl. Probab., 26(2):643–690, 2016

work page 2016

[28] [28]

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

work page 2007

[29] [29]

E. Powell. Critical Gaussian multiplicative chaos: a review.Markov Process. Related Fields, 27(4):557–506, 2021

work page 2021

[30] [30]

Saksman and C

E. Saksman and C. Webb. The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line.Ann. Probab., 48(6):2680–2754, 2020

work page 2020

[31] [31]

Soundararajan and M

K. Soundararajan and M. W. Xu. Central limit theorems for random multiplicative functions.J. Anal. Math., 151(1):343–374, 2023

work page 2023

[32] [32]

Soundararajan and A

K. Soundararajan and A. Zaman. A model problem for multiplicative chaos in number theory.Enseign. Math., 68(3-4):307–340, 2022. Mathematical Institute, University of Oxford Email address:hamdan@maths.ox.ac.uk

work page 2022