A short proof of Helson's conjecture

Let α N → S¹ be the Steinhaus multiplicative function: a completely multiplicative function such that (α(p))p prime are i.i.d. random variables uniformly distributed on S¹. Helson conjectured that E|∑n≤ xα(n)|=o(√x) as x → ∞, and this was solved in strong form by Harper. We give a short proof of the conjecture using a result of Saksman and Webb on a random model for the Riemann zeta function.