Almost Sure Large Fluctuations of Random Multiplicative Functions

Abstract We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum _{n \leq x} f(n)| \geq \sqrt{x} (\log \log x)^{1/4+o(1)}$. This the first such bound grows faster than $\sqrt{x}$, answering question Halász and proving conjecture Erd?s. It plausible exponent $1/4$ sharp in this problem. The proofs work by establishing multivariate Gaussian approximation sums $\sum f(n)$ at sequence $x$, conditional on behaviour $f(p)$ all except largest primes $p$. most difficult aspect showing covariances are usually small, so corresponding Gaussians roughly independent. These related to Euler product (or chaos) type integral twisted additive characters, we study using various tools including mean value estimates Dirichlet polynomials, high mixed moment products, barrier arguments with walks.