Helson's problem for sums of a random multiplicative function

TL;DR

This paper investigates the behavior of sums of random multiplicative functions, establishing lower bounds on their expected magnitudes and moments, which advances understanding of their probabilistic properties.

Contribution

It provides new lower bounds on the expected size and moments of sums of random multiplicative functions, extending previous results in probabilistic number theory.

Findings

Expected sum magnitude grows at least as ^{0.5} with a logarithmic correction.

Moments of the sum also exhibit similar growth with specific logarithmic factors.

Results improve understanding of the size and distribution of random multiplicative sums.

Abstract

We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.