A short proof of Helson's conjecture

Let alpha:N -> S1$\alpha \colon \mathbb {N}\rightarrow S<^>1$ be the Steinhaus multiplicative function: a completely multiplicative function such that (alpha(p))pprime$(\alpha (p))_{p\text{ prime}}$ are i.i.d. random variables uniformly distributed on the complex unit circle S1$S<^>1$. Helson conjectured that E|& sum;n <= x alpha(n)|=o(x)$\mathbb {E}|\sum _{n\leqslant x}\alpha (n)|=o(\sqrt {x})$ as x ->infinity$x \rightarrow \infty$, and this was solved in a 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 zeta function.