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Moments of random multiplicative functions with polynomial coefficients

Xinyu Wang

math.NT Sep 28, 2026 · v1
formalization number-theory probability
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TL;DR
Lean 4 code is provided to verify the proofs of the moment dichotomy results for random multiplicative functions.
Abstract
Let $f$ be a Steinhaus random multiplicative function and let $g$ be a polynomial of degree $d$. Write $S_N=N^{-1/2}\sum_{n\le N}f(n)\,e(g(n))$. We prove a quantitative dichotomy for the moments of $S_N$: for each integer $s\ge 2$, either $\mathbb{E}|S_N|^{2s}$ is close to the Gaussian moment $s!$, or the coefficients of $g$ can be approximated by rationals with a small denominator. In particular, if the coefficients of $g$ satisfy a Diophantine condition, then $S_N$ converges in law to a complex normal distribution with mean $0$ and variance $1$. Lean 4 code for the proofs is provided, for convenience of verification.
Problem

For a Steinhaus random multiplicative function f and a real polynomial g of degree d, one studies the moments of the normalized sum S_N = N^{-1/2} sum_{n<=N} f(n) e(g(n)). The question is how the Diophantine properties of the polynomial coefficients control the moment behaviour.

Approach

The 2s-th moment is reduced via orthogonality to an exponential sum over the multiplicative solution set V of n_1...n_s = m_1...m_s. An inclusion-exclusion decomposition of V using large-gcd sets is combined with divisor bounds and quantitative Green-Tao equidistribution/recurrence results to isolate main and error terms. The main theorem is proved by induction on s. Lean 4 code is supplied for verification of the proofs.

Results

A quantitative dichotomy is established: for each s>=2, either E|S_N|^{2s} is close to the Gaussian moment s!, or the coefficients of g are well-approximated by rationals with small denominator. Under a Diophantine condition, S_N converges in law to a standard complex normal distribution.