Helson's problem for sums of a random multiplicative function

Abstract

We consider the random functions $S_{N}(z):=\sum _{n=1}^{N}z(n)$SN(z):=∑Nn=1z(n), where $z(n)$z(n) is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$z(p). It is shown that $\mathbb{E}|S_{N}|\gg \sqrt{N}(\log N)^{-0.05616}$E|SN|≫N−−√(logN)−0.05616 and that $(\mathbb{E}|S_{N}|^{q})^{1/q}\gg _{q}\sqrt{N}(\log N)^{-0.07672}$(E|SN|q)1/q≫qN−−√(logN)−0.07672 for all $q>0$q>0