Year: 2026
Type: preprint
Abstract: Let $k\ge1$ be an integer and let $λ$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $λ(n+h_1),\dots, λ(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. Building on their work, we prove a dynamical generalization of Tao's theorem and a variant for the composition of the sum-of-digits function with the prime Omega function.
Source: arXiv (Cornell University)
Author: Wang, Biao
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8974978213
OpenAlex ID: W7203715718