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We give lower bounds for the small moments of the sum of a randommultiplicative function, which improve on some results of Bondarenko and Seipand constitute further progress towards (dis)proving a conjecture of Helson. Wealso prove asymptotics for the even integer moments. The latter have also beenobtained very recently and independently by Heap and Lindqvist. Our proofsinvolve general lower bound techniques for random multiplicative functions,mean value results for multiplicative functions in several variables, and somecalculations with Birkhoff polytopes.

A note on Helson's conjecture on moments of random multiplicative functions