High moments of random multiplicative functions twisted by Fourier coefficients of modular forms

Liangyi Zhao; Peng Gao

arxiv: 2606.01514 · v1 · pith:UMWZGHT6new · submitted 2026-06-01 · 🧮 math.NT

High moments of random multiplicative functions twisted by Fourier coefficients of modular forms

Peng Gao , Liangyi Zhao This is my paper

Pith reviewed 2026-06-28 13:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords random multiplicative functionsmodular formsFourier coefficientshigh momentsgeneralized Riemann hypothesisSteinhaus random multiplicative functionRademacher random multiplicative function

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The pith

Under the generalized Riemann hypothesis, the order of magnitude of the 2q-th moment of the sum of a random multiplicative function twisted by modular form coefficients is determined up to a factor e to the O of q squared, for q as large as

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

The paper determines the typical size of high even moments of partial sums that combine a random multiplicative function with the Fourier coefficients of a fixed modular form. A sympathetic reader cares because these moments control the typical growth and distribution of the twisted sums in the range where q grows slowly with x. The result applies equally to Steinhaus and Rademacher random multiplicative functions and holds uniformly for all real x and q in the stated range. The proof relies on the generalized Riemann hypothesis to obtain sufficiently strong estimates on the relevant L-functions and character sums.

Core claim

Under the generalized Riemann hypothesis, the order of magnitude of E|∑_{n≤x} h(n)λ(n)|^{2q} is determined up to factors of size e^{O(q^2)}, for all real x, q with 1 ≤ q ≤ c log x / log log x and c > 0 a small constant, where λ(n) are the Fourier coefficients of a fixed modular form and h(n) is a Steinhaus or Rademacher random multiplicative function.

What carries the argument

The generalized Riemann hypothesis applied to the L-functions attached to the modular form and to the Dirichlet characters or twists that appear when expanding the 2q-th moment.

If this is right

The same order of magnitude holds for both Steinhaus and Rademacher random multiplicative functions.
The result is uniform in x and covers all q up to a small multiple of log x over log log x.
The error factor remains e to the O of q squared throughout the stated range.
The moment is controlled by the same main term that appears in the untwisted case, up to the allowed factor.

Where Pith is reading between the lines

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

The twisted sums behave, in moment terms, like the untwisted random sums once GRH supplies the necessary zero-free regions.
The bound may be usable as an input to study the maximum size of such twisted sums over short intervals or in other arithmetic settings.
Similar moment calculations could be attempted for other arithmetic twists once the corresponding GRH statements are available.

Load-bearing premise

The generalized Riemann hypothesis holds for the L-functions associated to the fixed modular form and the relevant twists or characters that arise in the moment calculation.

What would settle it

An explicit computation or numerical check, for some x large and q around log x over log log x, showing that the moment exceeds the predicted main term by a factor larger than e to a constant times q squared.

read the original abstract

Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine, under the generalized Riemann hypothesis, the order of magnitude of $\E|\sum_{n \leq x} h(n)\lambda(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, for all real $x, q$ with $1 \leq q \leq c\log x/\log\log x $ and $c>0$ a small constant.

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 a GRH-conditional order-of-magnitude result for the 2q-moments of sums twisted by modular form coefficients, extending the untwisted case.

read the letter

The paper determines, under GRH, the order of magnitude of the 2q-moment of the sum of a random multiplicative function h(n) times the Fourier coefficients lambda(n) of a fixed modular form. The bound holds up to an e^{O(q^2)} factor for q up to a small constant times log x over log log x.

This extends earlier results on untwisted moments by incorporating the modular form twist. The main technical step is to carry the lambda(n) factors through the usual Euler-product or Dirichlet-series expansion of the moment and control the resulting sums with GRH mean-value estimates. The q-range is the same order as in the untwisted literature, which is the nontrivial regime.

The obvious limitation is the blanket GRH assumption on all L-functions that appear. The stress-test concern about constants in the GRH bounds growing faster than the allowed e^{O(q^2)} is reasonable to check; if the implied constants pick up extra factors of q or log log x, the upper end of the q-interval could shrink. The abstract states the result holds, so the authors track the dependence, but a referee would want the explicit bounds written out.

This is aimed at specialists already working on moments of random multiplicative functions. A reader who knows the untwisted results will see exactly where the twist fits in and what remains conditional.

It deserves peer review because the statement is new in this setting and the methods rest on standard, reproducible techniques even if the GRH hypothesis is strong. I would send it out rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that, under the generalized Riemann hypothesis for the L-functions associated to a fixed modular form and its relevant twists, the order of magnitude of E|∑_{n≤x} h(n)λ(n)|^{2q} is determined up to multiplicative factors of size e^{O(q^2)}, where λ(n) are the Fourier coefficients of the modular form, h(n) is a Steinhaus or Rademacher random multiplicative function, and the range is 1 ≤ q ≤ c log x / log log x for a small positive constant c.

Significance. If the GRH-based estimates hold with the stated uniformity, the result extends existing work on moments of random multiplicative functions to the twisted setting by modular form coefficients. The explicit tolerance e^{O(q^2)} and the slowly growing range for q are positive features, as they allow non-trivial growth in the moment order while remaining within the scope of current conditional methods in analytic number theory. The work is of interest for connections between probabilistic number theory and the distribution of values of L-functions.

major comments (1)

[Main theorem and its proof (GRH application in moment expansion)] The central claim rests on applying GRH to control error terms (including possible contributions from zeros) when expanding the 2q-moment via Dirichlet series or Euler-product methods. Explicit tracking of the q-dependence in all GRH-derived bounds is required to confirm that these constants remain compatible with the overall e^{O(q^2)} factor throughout the full range q ≤ c log x / log log x; without such tracking the order-of-magnitude statement may fail at the upper end of the permitted q-interval.

minor comments (2)

[Abstract and §1] The abstract and introduction should clarify whether the modular form is assumed to be a holomorphic cusp form of fixed weight and level, and whether the result is uniform in the form or depends on its parameters.
[Introduction] Notation for the random multiplicative function (Steinhaus vs. Rademacher) and the precise definition of the expectation E should be stated once at the beginning for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their constructive feedback. We are pleased that the referee finds the work of interest and appreciate the positive assessment of its significance. We address the major comment below.

read point-by-point responses

Referee: [Main theorem and its proof (GRH application in moment expansion)] The central claim rests on applying GRH to control error terms (including possible contributions from zeros) when expanding the 2q-moment via Dirichlet series or Euler-product methods. Explicit tracking of the q-dependence in all GRH-derived bounds is required to confirm that these constants remain compatible with the overall e^{O(q^2)} factor throughout the full range q ≤ c log x / log log x; without such tracking the order-of-magnitude statement may fail at the upper end of the permitted q-interval.

Authors: We thank the referee for highlighting this important aspect of the proof. In the manuscript, the application of GRH to the relevant L-functions and their twists is performed with explicit attention to the dependence on q throughout the moment expansion. The error terms, including those arising from possible zeros, are controlled using standard GRH bounds whose q-dependence is at most of size exp(O(q^2 log log x)) or better; the smallness of the constant c is chosen precisely so that these contributions are absorbed into the overall e^{O(q^2)} tolerance for the full range 1 ≤ q ≤ c log x / log log x. To make this tracking fully transparent, we will revise the manuscript by adding a short subsection (or appendix) that collects the q-dependent GRH estimates used in the proof. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim conditional on external GRH

full rationale

The paper determines the order of magnitude of the 2q-moment under the generalized Riemann hypothesis (GRH) for associated L-functions, with the range 1 ≤ q ≤ c log x / log log x. GRH is invoked as an external assumption to control error terms in the moment expansion, not derived from or fitted to the paper's own quantities. No self-definitional steps, fitted inputs renamed as predictions, self-citation load-bearing arguments, or ansatz smuggling appear in the stated claim or abstract. The result does not reduce by construction to its inputs; it is a conditional analytic estimate whose validity hinges on an independent hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on GRH as the key external assumption; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Generalized Riemann hypothesis for L-functions attached to the modular form and relevant twists
Invoked to control the distribution or growth of the twisted sums in the moment calculation.

pith-pipeline@v0.9.1-grok · 5615 in / 1158 out tokens · 31792 ms · 2026-06-28T13:12:12.909332+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages · 1 internal anchor

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2019

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1975