Jad Hamdan

A simple proof of Helson's conjecture and some generalisations

Let (Z_p)_p be a sequence of independent, identically distributed random variables indexed by the primes, each uniformly distributed on the complex unit circle. A Steinhaus random multiplicative function f:N -->C is defined by setting f(p) = Z_p for each prime p, and extending completely multiplicatively to all positive integers.
A conjecture of Helson, proved by Harper, states that partial sums of f typically exhibit better than square-root cancellation. I will present a simple proof of this result, along with some extensions that can be obtained using the same approach.