On Equal Consecutive Values of Multiplicative Functions

Mangerel, Alexander P

Abstract

Let f : N → C be a multiplicative function for which ∑ p: | f (p)|̸ =1 1 p = ∞. We show under this condition alone that for any integer h ̸ = 0 the set {n ∈ N : f (n) = f (n + h) ̸ = 0} has logarithmic density 0. We also prove a converse result, along with an application to the Fourier coefficients of holomorphic cusp forms. The proof involves analysing the value distribution of f using the compositions | f | it , relying crucially on various applications of Tao's theorem on logarithmically-averaged correlations of non-pretentious multiplicative functions. Further key inputs arise from the inverse theory of sumsets in continuous additive combinatorics.

Citation

Mangerel, A. P. (2024). On Equal Consecutive Values of Multiplicative Functions. Discrete Analysis, 12, https://doi.org/10.19086/da.125450

• Journal Article Type: Article
• Acceptance Date: Jul 19, 2024
• Online Publication Date: Nov 11, 2024
• Publication Date: Nov 11, 2024
• Deposit Date: Jul 24, 2024
• Publicly Available Date: Nov 20, 2024
• Journal: Discrete Analysis
• Electronic ISSN: 2397-3129
• Peer Reviewed: Peer Reviewed
• Volume: 12
• DOI: https://doi.org/10.19086/da.125450.
• Public URL: https://durham-repository.worktribe.com/output/2602311
• Publisher URL: https://discreteanalysisjournal.com/
• Other Repo URL: https://doi.org/10.48550/arXiv.2306.09929