Oleksiy Klurman

On the random Chowla conjecture

A celebrated conjecture of Chowla in analytic number theory asserts that for the Liouville function $\lambda (n)$ and any non-square polynomial $P(x)\in\mathbb{Z}[x]$ one expects cancellations $\sum_{n\le x}\lambda(P(n))=o(x).$ In the case $P(n)=n,$ this corresponds to the prime number theorem, but the conjecture is widely open for any polynomial of $\deg P\ge 2.$ In 1944, Wintner proposed to study random model for this question (in the case $P(n)=n$) where $\lambda (n)$ is replaced by a random multiplicative function $f(n).$ The aim of the talk is to discuss recent advances in understanding the distribution and the size of the largest fluctuations of appropriately normalized partial sums $\sum_{n\le x}f(n)$ (mostly due to Harper) and my recent joint work with I. Shkredov and M. W. Xu aiming to understand $\sum_{n\le x}f(P(n))$ for any polynomial of $\deg P\ge 2.$