Maximum values of multiplicative functions on short intervals (Q6619685)

Let \(f\) be a multiplicative function satisfying some certain conditions, including \(|f(p)|\leq B\) for all primes \(p\leq X\), for some \(B\geq 1\). For any fixed integer \(\ell\geq 2\), let\N\[\Nf_\ell(n):=\max_{0\leq j\leq \ell-1}\left\{f(n+j)\right\}.\N\]\NIn the paper under review, the authors first prove an estimate of the following form\N\[\N\sum_{n\leq X}f_\ell(n)=\ell\sum_{n\leq X}f(n)+O_{f, \ell}\left(X\prod_{p\leq X}\left(1+\frac{\sqrt{f(p)-1}}{p}\right)^2\right),\N\]\Nwith corollaries to generalized divisor function and Hecke eigenvalues of a primitive holomorphic cusp form of even integral weight. Then, they give analogue result to the breakthrough work of \textit{K. Matomäki} and \textit{M. Radziwiłł} [Ann. Math. (2) 183, No. 3, 1015--1056 (2016; Zbl 1339.11084)] relating the above average in long intervals of the form \([X, 2X]\), to average of \(f_\ell(n)\) in short intervals of length \(h(\log X)^c\).