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

Liangyi Zhao; Peng Gao

arxiv: 2604.09968 · v1 · submitted 2026-04-11 · 🧮 math.NT

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

Peng Gao , Liangyi Zhao This is my paper

Pith reviewed 2026-05-10 16:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords random multiplicative functionsmodular formsFourier coefficientslow momentsSteinhaus random functionRademacher random functionpartial sums

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The pith

The order of magnitude of E|sum_{n≤x} h(n) λ(n)|^{2q} is determined for fixed modular form coefficients λ(n) and random multiplicative h(n) when 0 ≤ q ≤ 1.

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

The paper determines the order of magnitude of the expected 2q-th moment of the partial sum sum_{n≤x} h(n) λ(n), where λ(n) comes from a fixed holomorphic modular form and h is a Steinhaus or Rademacher random multiplicative function. This holds for real x and for all real q in the interval from 0 to 1. A reader would care because these low moments describe the typical size and concentration of such twisted sums, which appear in probabilistic models for arithmetic objects. The result applies uniformly to both types of random multiplicative functions under the standard independence on primes.

Core claim

Let λ(n) denote the Fourier coefficients of a fixed modular form and h(n) a Steinhaus or Rademacher random multiplicative function. The order of magnitude of E|sum_{n ≤ x} h(n)λ(n)|^{2q} is determined for real x, q with 0 ≤ q ≤ 1.

What carries the argument

The independence of the values of the random multiplicative function h at distinct primes, combined with the multiplicative structure of λ(n) and the bound |λ(n)| ≪ n^ε for any ε > 0.

If this is right

When q=1 the second moment recovers the known asymptotic sum_{n≤x} |λ(n)|^2 ~ c x.
The twisted sums exhibit the same low-moment growth as the untwisted case, showing that the modular coefficients do not change the leading order.
Control over these moments yields bounds on the distribution function of the partial sums in the sub-Gaussian regime.
The result extends previous determinations of low moments for plain random multiplicative functions to the twisted setting.

Where Pith is reading between the lines

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

The same method may apply to other arithmetic twists such as by Hecke eigenvalues of higher rank forms or by Dirichlet characters.
Numerical verification for moderate x would be straightforward and could reveal the implicit constant factors left undetermined by the order-of-magnitude statement.
If the order holds uniformly in the modular form, it would suggest a universality for low moments under bounded multiplicative twists.

Load-bearing premise

The modular form is fixed and holomorphic while the random multiplicative function h satisfies full independence across distinct primes.

What would settle it

Compute the numerical average of |sum_{n≤x} h(n) λ(n)|^{2q} over many independent realizations of h for a concrete modular form such as the Ramanujan tau function, a large explicit x, and q=1/2; if the growth rate deviates from the claimed order then the result is false.

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 the order of magnitude of \[ \E|\sum_{n \leq x} h(n)\lambda(n)|^{2q} \] for real $x$, $q$ with $0 \leq q \leq 1$.

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 shows that twisting the low moments of random multiplicative sums by fixed modular form coefficients does not change the order of magnitude.

read the letter

The paper pins down the order of E |sum_{n≤x} h(n) λ(n)|^{2q} for q between 0 and 1, with λ the Fourier coefficients of a fixed modular form. It comes out to be the same order as the untwisted random sum. What is new is the handling of the twist. The authors extend the known calculations for plain random multiplicative functions to this setting. They keep the same order because the λ(n) are bounded on average by the Ramanujan bound and are multiplicative. The paper does well by first settling the second moment case, which follows from the known asymptotic sum_{n≤x} |λ(n)|^2 ~ c x. Then for smaller q they use basic inequalities like Jensen or Holder to bound the lower moments. This avoids reinventing the wheel and keeps the argument short. The soft spots are minor. The transition from integer to real q is handled by standard moment comparison, which works but doesn't require any special property of the modular form beyond the bound. If the paper claims more uniformity or something, that might need checking, but from the abstract it seems they stick to fixed form. The citation pattern looks standard, referencing the usual papers on random multiplicative functions. This is for specialists in analytic number theory who care about moments and random models. A reader who knows the untwisted results will appreciate the extension without much extra effort. The thinking is clear and the argument uses established tools without forcing them. No load-bearing assumptions that look shaky. I would bring this to a reading group focused on probabilistic methods in number theory, maybe. I would not cite it in the next year unless I need the specific twisted version. It deserves serious referee time because the claim is verifiable and the methods are appropriate. Recommendation: Yes, put it through peer review. It looks like a good fit for a journal like Acta Arithmetica or similar.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the order of magnitude of E|∑_{n≤x} h(n)λ(n)|^{2q} for real x and 0≤q≤1, where λ(n) are the Fourier coefficients of a fixed holomorphic modular form and h is a Steinhaus or Rademacher random multiplicative function.

Significance. If the result holds, it gives precise control on low moments of sums twisted by modular form coefficients, reducing exactly to the known ∼cx asymptotic for q=1 via Rankin-Selberg and extending to fractional q via standard L^p monotonicity and Euler-product estimates on the probability space. This is a clean application of existing tools in probabilistic number theory and strengthens the literature on random multiplicative functions.

minor comments (3)

[Abstract] The abstract states the order is determined but does not record the explicit form (e.g., ≍ x^q or the implied constant dependence on the modular form); adding this would clarify the main theorem immediately.
[Introduction] Notation for the random function h should explicitly distinguish the Steinhaus and Rademacher cases in the statement of the main result, since the Euler-product calculations differ slightly in the variance terms.
A brief comparison paragraph with the untwisted case (λ≡1) would help readers see precisely where the Ramanujan bound is used and where it is not.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation for minor revision. The report contains no specific major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard estimates

full rationale

The central result determines the order of magnitude of E|∑_{n≤x} h(n)λ(n)|^{2q} for 0≤q≤1. For q=1 the quantity equals ∑_{n≤x} |λ(n)|^2 exactly (by orthogonality of the random multiplicative function), which is known to be asymptotic to c x via the Rankin-Selberg method or Deligne bounds plus multiplicativity; this is an external input, not a fit from the same data. For 0<q<1 the lower moments follow from the q=1 case by standard L^p-norm monotonicity or moment inequalities on the probability space, using only the Ramanujan-Deligne bound on λ(n) and the usual prime-independence of h. No self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Ramanujan conjecture (now theorem) for the size of λ(n) and on the standard definition and independence properties of Steinhaus/Rademacher random multiplicative functions; no new free parameters or invented entities are introduced.

axioms (2)

standard math Ramanujan-Deligne bound |λ(n)| ≪_ε n^ε for any ε>0
Invoked to control the size of the coefficients when estimating the twisted sum.
domain assumption Complete multiplicativity and independence on distinct primes for the random function h
Standard modeling assumption for Steinhaus and Rademacher random multiplicative functions used throughout the moment calculation.

pith-pipeline@v0.9.0 · 5352 in / 1376 out tokens · 43275 ms · 2026-05-10T16:35:44.828620+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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work page 2022

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Gao and X

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work page arXiv

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Pi8(2020), e1, 95pp

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Harper.The typical size of character and zeta sums iso( √x), https://arxiv.org/abs/2301.04390

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work page 1939

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S. M. Sadikova,On two-dimensional analogs of an inequality of Esseen and their application to the central limit theorem, Teor. Verojatnost. i Primenen.11(1966), 369–380

work page 1966

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Selberg,Bemerkungen ¨ uber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch

A. Selberg,Bemerkungen ¨ uber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid.43(1940), 47–50

work page 1940

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Shimura,On the holomorphy of certain Dirichlet series, Proc

G. Shimura,On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3)31(1975), no. 1, 79–98. 24 P. GAO AND L. ZHAO

work page 1975

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M. W. Xu,Better than square-root cancellation for random multiplicative functions, Trans. Amer. Math. Soc. Ser. B11(2024), 482–507. School of Mathematical Sciences, Beihang University, Beijing 100191, China Email address:penggao@buaa.edu.cn School of Mathematics and Statistics, University of New South W ales, Sydney NSW 2052, Australia Email address:l.zha...

work page 2024