On moments of gaps between consecutive squarefree numbers

Abstract

Let $s_1,s_2,s_3,\cdots$ be the set of squarefree numbers in ascending order. In this paper, we prove that the following asymptotic on moments of gaps between squarefree numbers \[ \sum_{s_{k+1} \le x} (s_{k+1} - s_k)^γ\sim B(γ) x \; \; \mbox{ with some constant} \; \; B(γ) > 0 \] is true for $0\le \gamma <3.75$. This improves the previous best range $0\le \gamma <3.6875$.

10 pages