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

Abstract

AbstractWith 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 $${\varvec{g:}} \mathbb {N} \rightarrow \mathbb {C}$$ g : N → C any non-trivial savings in the size of the average gap $$|{\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}-{\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}} |$$ | g ( n ) - g ( n - 1 ) | implies that $${\varvec{g}}$$ g must have a small first centred moment i.e. the discrepancy of $${\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}$$ g ( n ) 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 $${{\textbf {log}}} \,{\varvec{n}}$$ log n 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 $$\{{\varvec{n}} \in \mathbb {N} : {\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}} < {\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}}\}$$ { n ∈ N : g ( n ) < g ( n - 1 ) } is sufficiently sparse, and if $${\varvec{g}}$$ g is not extremely large too often on the primes (in a precise sense), then $${\varvec{g}}$$ g is identically equal to a constant times a logarithm.