On the Hardy–Littlewood–Chowla conjecture on average

Journal article

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any k,l >= 1 and distinct integers h(2), ..., h(k), a(1), ...., a(l), we have:Sigma(n for all except o(H) values of h(1) = (log X) (l+epsilon). This improves on the range H >= (log X)(psi (X)) , psi(X) -> infinity, obtained in previous work of the first author. Our results also generalise from the Mobius function mu to arbitrary (non-pretentious) multiplicative functions.

Authors: Lichtman, JD; Teräväinen, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Forum of Mathematics, Sigma
• Volume: 10
• Pages: e57
• Article number: e57
• Publication date: 2022-07-27
• DOI: 10.1017/fms.2022.54
• EISSN: 2050-5094
• ISSN: 2050-5094
• Language: English
• Keywords: Physics; Mathematics; Quantum mechanics; Conjecture; Mathematical analysis; Multiplicative function; Lambda; Combinatorics