  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.
