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Moments of random multiplicative functions with polynomial coefficients
Abstract
Let f be a Steinhaus random multiplicative function and let g be a polynomial of degree d. Write SN=N−1/2∑n≤ Nf(n) e(g(n)). We prove a quantitative dichotomy for the moments of SN: for each integer s≥ 2, either 𝔼|SN|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 SN 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.
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BibTeX
@misc{wang2026moments,
title = {{Moments of random multiplicative functions with polynomial coefficients}},
author = {Xinyu Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35187},
eprint = {2609.35187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35187},
}
RIS
TY - UNPB
TI - Moments of random multiplicative functions with polynomial coefficients
AU - Xinyu Wang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35187
UR - https://arxiv.org/abs/2609.35187
AB - Let f be a Steinhaus random multiplicative function and let g be a polynomial of degree d. Write S_N=N^−1/2∑_n≤ Nf(n) e(g(n)). We prove a quantitative dichotomy for the moments of S_N: for each integer s≥ 2, either 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.
ER -