Understanding joint behaviour of $(f(n),g(n+1))$ where f and g are given multiplicative functions play key role in analytic number theory with potentially profound consequences such as Riemann hypothesis, twin prime conjecture, Chowla's conjecture and many others.

In the the first part of this talk, I will discuss joint work with A. Mangerel, answering an old question of Katai about distribution of points $\{(f(n),g(n+1))\}_{n\ge 1}\in \mathbb{T}^2,$ where f and g are unimodular multiplicative functions.  

In the second part of the talk, which is based on a joint work with P. Kurlberg, answering a question of M. Lemanczyk, we construct deterministic example of multiplicative function $f:{\mathbb{N}\to \{+1,-1\}$ with various ergodic properties with respect to the Mirsky measure and discuss its relation to the interplay between Chowla conjecture and Riemann hypothesis.