Helson's problem for sums of a random multiplicative function

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 ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$andthat$({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$forall$q>0$.