Subconvexity of Short $k$-Free Exponential Sums

Abstract

Let $S_k(\alpha ;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha ;K)$ whenever $K\gg N^{{\theta }_{k,s}+\epsilon }$ for some ${\theta }_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the Möbius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an ${\ell }^2$-estimate involving the Möbius function.

27 pages