Oleksiy Klurman (University of Bristol)

On the random Chowla conjecture

Abstract:

A celebrated conjecture of Chowla in number theory asserts that for the Liouville function λ(n) and any non-square polynomial P(x)∈ℤ[x] one expects cancellations ∑_n≤x λ(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 degree deg(P) ≥ 2. In 1944, Wintner proposed to study random model for this question (in the case P(n)=n) where λ(n) is replaced by a random multiplicative function f(n). Since then, this problem attracted a lot of attention in probability, number theory and analytic communities. 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 ∑_n≤x f(n)$ (mostly due to Harper) and my recent joint work with I. Shkredov and M. W. Xu aiming to understand ∑_n≤x f(P(n)) for any polynomial of deg(P)≥ 2.