### Furstenberg systems of multiplicative functions and beyond

Ergodic Theory and Dynamical Systems Seminar

9th March 2023, 2:00 pm – 3:00 pm

Fry Building, G07

A well-known conjecture of Sarnak predicts that each zero-entropy continuous map T of a compact metric space X is Möbius disjoint, that is

$\lim_{x\to \infty} \frac{1}{x}\sum_{n\le x} f(T^nx)\mu(n)=0$ for every continuous $f$ and $x\in X.$ This is still open in general, but spectacular progress has been made recently in part due to advances in our understanding of autocorrelations of the Möbius function and of the so-called Furstenberg systems of bounded multiplicative functions.

In this talk, i will present a general way of constructing multiplicative functions with small autocorrelations. This leads to several consequences including:

Construction of multiplicative functions with a given Furstenberg system, thus answering a question of Lemanczyk.

A proof that Chowla's property does not imply the Riemann property, i.e there are multiplicative functions with small autocorrelations that do not exhibit square-root cancellation of partial sums (the object of some recent speculation).

Construction of multiplicative semigroup of $\mathbb{N}$ with Poissonian gaps statistic, thus providing an (unconditional) multiplicative analogue of the classical result of Gallagher about the primes.

This is joint work with P. Kurlberg, S. Mangerel and J.Teräväinen.

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