Maximum values of multiplicative functions on short intervals
Tom 215 / 2024
Acta Arithmetica 215 (2024), 229-247
MSC: Primary 11F30; Secondary 11N37
DOI: 10.4064/aa230611-12-3
Opublikowany online: 30 August 2024
Streszczenie
Let be a non-negative multiplicative function which is uniformly bounded. In this paper, we study asymptotics of the maximum values of f(n) on intervals of length l, i.e. of the function f_{l}(n):= \max\{f(n) , f(n+1) , \ldots , f(n+l - 1)\}. We first establish an asymptotic formula for the summatory function of f_{l}(n) over long-range n. Our main aim is to show that this formula persists in typical short intervals. To this end, we give uniform upper bounds for the variance of averages of f_{l}(n) over intervals of length h(\log X)^c, with c \gt 0 explicit, as h=h(X)\rightarrow \infty . We present two applications of this result to higher order divisor functions and Hecke eigenvalues.
