Dynamical Chowla conjecture for polynomials of several variables

Abstract

In 1965, Chowla conjectured that the mean value of the Liouville function at non-square one-variable polynomials is zero. This conjecture is a generalization of the prime number theorem (PNT). In 2022, Bergelson and Richter established a dynamical generalization of this form of the PNT. For non-square multivariate polynomials, one also expects that Chowla's conjecture is true. In this article, we extend Bergelson and Richter's result to multivariate polynomials whose Birch rank is sufficiently large in terms of their degree. We also establish analogous results for averages over integer points on algebraic varieties.