Year: 2026
Type: preprint
Abstract: In 1965, Chowla conjectured that the signs of the Liouville function become asymptotically uncorrelated at any fixed collection of distinct shifts. In 2015, Matomäki, Radziwiłłand Tao proved an averaged form of Chowla's conjecture. In 2022, Lichtman proved a variant of this conjecture over primes on average. In the same year, Lichtman and Teräväinen proved the Hardy-Littlewood-Chowla conjecture on average. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will first establish the analogous dynamical generalizations of these three results related to Chowla's conjecture. In the proofs, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. 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 will also prove a dynamical generalization of Tao's theorem and some variants for the composition of the sum-of-digits function with the prime Omega function.
Source: arXiv (Cornell University)
Authors: Wang, Biao, Yi, Shaoyun
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213333870