A note on Helson's conjecture on moments of random multiplicative functions

TL;DR

This paper advances understanding of Helson's conjecture by providing improved lower bounds for small moments of random multiplicative functions and establishing asymptotics for even moments, using novel mathematical techniques.

Contribution

It introduces new lower bound methods and asymptotic results for moments of random multiplicative functions, progressing towards resolving Helson's conjecture.

Findings

Improved lower bounds for small moments

Asymptotic formulas for even moments

Application of Birkhoff polytope calculations

Abstract

We give lower bounds for the small moments of the sum of a random multiplicative function, which improve on some results of Bondarenko and Seip and constitute further progress towards (dis)proving a conjecture of Helson. We also prove asymptotics for the even integer moments. The latter have also been obtained very recently and independently by Heap and Lindqvist. Our proofs involve general lower bound techniques for random multiplicative functions, mean value results for multiplicative functions in several variables, and some calculations with Birkhoff polytopes.