Maximum values of multiplicative functions on short intervals

Let $f$ 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.