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arxiv: 2509.19067 · v4 · submitted 2025-09-23 · 🧮 math.PR · math.DS· math.NT

A few notes on the asymptotic behavior of Rademacher random multiplicative functions

Yeor Hafouta This is my paper

Pith reviewed 2026-05-18 14:23 UTC · model grok-4.3

classification 🧮 math.PR math.DSmath.NT
keywords Rademacher random multiplicative functionspartial sumshigh momentsmartingaleBurkholder inequalityasymptotic behaviorconcentration inequalities
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The pith

Explicit bounds on high moments of partial sums S_n for Rademacher random multiplicative functions hold without restricting moment order relative to n.

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

The paper examines the sums S_n formed from products of independent random signs X_p attached to square-free integers. It derives moment estimates that apply for any moment order, no matter how large compared to n, by treating the partial sums as a martingale and invoking Burkholder's inequality. These bounds are used to study possible normalizations a_n under which S_n / a_n might converge, extending earlier results that already rule out convergence after division by sqrt(n) or by slightly smaller sequences.

Core claim

Viewing S_n as a martingale with respect to the filtration generated by the primes up to the current index, the Burkholder inequality yields explicit constants C_p such that the p-th moment of S_n is bounded by C_p times n^{p/2} times a slowly varying factor that remains controlled even when p grows with or exceeds n.

What carries the argument

Martingale structure of the partial sums S_n together with the Burkholder inequality applied to the increments driven by each new prime.

If this is right

  • The moment bounds immediately give exponential concentration inequalities for S_n in the large-deviation regime.
  • The same estimates apply to a natural combinatorial counting problem in number theory that can be recast as an expectation involving these sums.
  • The bounds remove the previous technical restriction that the moment order must be smaller than n, allowing broader use in limit theorems for multiplicative functions.

Where Pith is reading between the lines

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

  • Similar martingale-plus-Burkholder arguments might yield moment bounds for the Steinhaus (complex) case that were recently settled by other methods.
  • The explicit constants could be fed into existing truncation arguments to test whether any a_n = o(b_n) produces a non-degenerate limit for S_n.
  • If the bounds remain sharp for very large p, they would constrain the tail behavior of S_n more tightly than previous work.

Load-bearing premise

The signs X_p for distinct primes are independent and each equals plus or minus one with probability one half, so that the partial sums form a martingale.

What would settle it

A direct numerical check for moderate n and large p showing that the p-th moment of S_n grows faster than the explicit bound derived from Burkholder would disprove the moment claim.

read the original abstract

Let $X_p, p\in\cP$ be a sequence of independent random variables s.t. $\bbP(X_p=\pm 1)=1/2$. Let $\te_j=\prod_{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. The from this point of view proving limit theorems for $S_n$ is natural problem, since $S_n$ mimics the behavior of $e^{\sqrt{\ln(\beta)}}$. It is a natural guiding conjecture that $S_n/\sqrt n$ obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite{[25]}) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite{[17]}, and by now it is a direct consequence of a more recent breakthrough by Harper \cite{Har20} that $\frac{S_n}{b_n}\to 0$ in $L^1$, where $b_n=(n^{1/2}(\ln(\ln(n)))^{-1/4})u_n, u_n\to\infty$. In particular $S_n/\sqrt n\to 0$. Nevertheless, the question whether there exists a sequence $a_n=o(b_n)$ such that $S_n/a_n$ converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by \cite{Gor1}. In this paper make an attempt to shed some light on the convergence of $S_n/a_n$. Additionally, we obtain explicit estimates on hight moments of $S_n$ without restrictions on the size of the moment compared to $n$ like in \cite[Theorem 1.2]{Har19}, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for $S_n$ (in the large deviations regime)

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

1 major / 2 minor

Summary. The manuscript studies the partial sums S_n = sum_{ℓ=1}^n ε_ℓ of Rademacher random multiplicative functions, where ε_ℓ = product_{p|ℓ} X_p for square-free ℓ (and 0 otherwise) with independent Rademacher X_p. It claims explicit estimates for high moments of S_n (of arbitrary order relative to n) via a martingale construction M_j = E[S_n | F_j] and Burkholder's inequality, without the restrictions appearing in prior combinatorial work. The paper also derives exponential concentration inequalities for S_n in the large-deviation regime and attempts to shed light on possible normalizing sequences a_n = o(b_n) for which S_n/a_n might converge.

Significance. If the claimed moment bounds hold without reintroducing size restrictions on the moment order versus n, the martingale-plus-Burkholder approach would supply a useful probabilistic tool for random multiplicative functions, complementing combinatorial methods and potentially aiding the open question on the existence of a_n for convergence of S_n/a_n. The independence of the X_p is used cleanly to form the martingale, and the application to a natural combinatorial problem is noted as a strength.

major comments (1)
  1. [Martingale/Burkholder moment section (around the application of Burkholder to the quadratic variation)] The central claim of explicit high-moment estimates without restrictions on moment size versus n (as contrasted with Har19 Thm 1.2) is load-bearing. In the section applying Burkholder's inequality to bound E[|S_n|^p] via E[[M,M]^{p/2}], the quadratic variation [M,M] = sum_j (Delta_j M)^2 consists of sums over square-free integers whose largest prime factor is the j-th prime. The manuscript must explicitly show how the p/2-moments of these sums are controlled without reintroducing a condition that p is not too large relative to n; otherwise the claimed improvement over prior work is not established.
minor comments (2)
  1. [Abstract] Abstract contains several grammatical and typographical issues: 'The from this point of view proving limit theorems for S_n is natural problem' should be rephrased; 'hight moments' should read 'high moments'; 'te_j' notation is introduced without clear motivation for the symbol choice.
  2. [Introduction] The discussion of the open question on a_n = o(b_n) for possible convergence of S_n/a_n is stated only at a high level; a more precise formulation of what 'shed some light' means (e.g., a specific conjecture or partial result) would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the moment estimates. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Martingale/Burkholder moment section (around the application of Burkholder to the quadratic variation)] The central claim of explicit high-moment estimates without restrictions on moment size versus n (as contrasted with Har19 Thm 1.2) is load-bearing. In the section applying Burkholder's inequality to bound E[|S_n|^p] via E[[M,M]^{p/2}], the quadratic variation [M,M] = sum_j (Delta_j M)^2 consists of sums over square-free integers whose largest prime factor is the j-th prime. The manuscript must explicitly show how the p/2-moments of these sums are controlled without reintroducing a condition that p is not too large relative to n; otherwise the claimed improvement over prior work is not established.

    Authors: We thank the referee for this observation, which correctly identifies that the claimed improvement requires careful justification at this step. The Burkholder inequality itself yields E[|S_n|^p] bounded in terms of E[[M,M]^{p/2}] with a constant depending only on p, and this holds for arbitrary p with no n-dependent restriction. However, we agree that the manuscript does not spell out in sufficient detail how the p/2-moments of the grouped sums (the increments Delta_j M, each a Rademacher multiplicative sum over integers with fixed largest prime factor p_j) are controlled. In the revised version we will insert an explicit lemma or subsection establishing these bounds via the independence of the X_p and the recursive multiplicative structure: each such grouped sum is distributed as a similar but smaller instance of the original problem (with reduced upper limit n/p_j), allowing the same martingale construction to be applied inductively on the prime index without introducing any auxiliary condition relating p and n. This makes the unrestricted nature of the high-moment estimates fully rigorous and establishes the improvement over the combinatorial restrictions in Har19. revision: yes

Circularity Check

0 steps flagged

No significant circularity; core estimates rely on external martingale and Burkholder tools

full rationale

The paper's claimed explicit high-moment estimates for S_n are presented as obtained via a martingale argument plus Burkholder's inequality, with the martingale property following directly from the stated independence of the X_p sequence. This is independent of the target bounds and does not reduce to a fitted parameter or self-referential definition. Citations to Harper (Har19, Har20) supply context and contrast but are not load-bearing for the new derivation; the paper explicitly contrasts its approach with the restrictions in Har19 Thm 1.2. No self-citation chains, ansatz smuggling, or renaming of known results appear in the load-bearing steps. The skeptic concern addresses potential correctness or completeness of the large-p regime but does not exhibit a quoted reduction of the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the independence of the prime signs and on two standard probabilistic tools; no new free parameters or postulated entities are introduced.

axioms (2)
  • domain assumption The random variables X_p for distinct primes p are independent with P(X_p = +1) = P(X_p = -1) = 1/2
    This independence is required to construct the martingale filtration and to apply Burkholder's inequality to the increments.
  • standard math Burkholder's inequality holds for the martingale constructed from the partial sums S_n
    Invoked to convert increment bounds into moment estimates of arbitrary order.

pith-pipeline@v0.9.0 · 5934 in / 1281 out tokens · 49349 ms · 2026-05-18T14:23:01.493861+00:00 · methodology

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