HELSON'S PROBLEM FOR SUMS OF A RANDOM MULTIPLICATIVE FUNCTION

We consider the random functions S-N(z) := Sigma(N)(n=1) (z(n)), where z(n) is the completely multiplicative random function generated by independent Steinhaus variables z(p). It is shown that (E vertical bar S-N vertical bar) >>(q) root N(log N)(-0.05616) and that (E vertical bar S-N vertical bar(q))(1/q) >>(q) root N(log N)(-0.07672) for all q > 0.