Additive functions in short intervals, gaps and a conjecture of Erdős

Mangerel, Alexander P.

Abstract

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function g:g:N→C any non-trivial savings in the size of the average gap |gg(nn)−gg(nn−1)| implies that gg must have a small first centred moment i.e. the discrepancy of gg(nn) from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of lognn as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set {nn∈N:gg(nn)

Citation

Mangerel, A. P. (2022). Additive functions in short intervals, gaps and a conjecture of Erdős. Ramanujan Journal, 59(4), 1023-1090. https://doi.org/10.1007/s11139-022-00623-y

• Journal Article Type: Article
• Acceptance Date: Jun 27, 2022
• Online Publication Date: Sep 3, 2022
• Publication Date: 2022-12
• Deposit Date: Sep 8, 2022
• Publicly Available Date: Mar 15, 2023
• Journal: The Ramanujan Journal
• Print ISSN: 1382-4090
• Electronic ISSN: 1572-9303
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 59
• Issue: 4
• Pages: 1023-1090
• DOI: https://doi.org/10.1007/s11139-022-00623-y
• Public URL: https://durham-repository.worktribe.com/output/1192217