Two-point correlations of multiplicative functions with dense orbits

Abstract

Let $f,g:\mathbb{N}\to \mathbb{T}$ be completely multiplicative functions with dense images in the complex unit circle $\mathbb{T}$. We prove that, for every non-empty open set $U\subset {\mathbb{T}}^2$, the set of integers $n$ such that $(f(n),g(n+1))\in U$ has positive upper logarithmic density, unless the pair $(f,g)$ is of a special form. This result strengthens earlier theorems of Klurman and Mangerel, as well as of Charamaras, Mountakis, and Tsinas, and yields simpler proofs.

arXiv commentThe hypothesis in the statement of Theorem 1.1 has been modified. We thank I. Kousek for pointing out a counterexample that led to this correction