Moments of random multiplicative functions and truncated characteristic polynomials

TL;DR

This paper derives asymptotic formulas for moments of sums of multiplicative Steinhaus variables and truncated characteristic polynomials of unitary matrices, establishing their equivalence and extending results to Rademacher variables.

Contribution

It provides new asymptotic formulas for moments of multiplicative functions and connects these to moments of characteristic polynomials, including Rademacher cases.

Findings

Asymptotic formulas for 2k-th moments of Steinhaus sums

Asymptotic equivalence with moments of truncated characteristic polynomials

Extension of results to multiplicative Rademacher variables

Abstract

We give an asymptotic formula for the $2k$th moment of a sum of multiplicative Steinhaus variables. This was recently computed independently by Harper, Nikeghbali and Radziwi\l\l. We also compute the $2k$th moment of a truncated characteristic polynomial of a unitary matrix. This provides an asymptotic equivalence with the moments of Steinhaus variables. Similar results for multiplicative Rademacher variables are given.