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 \[ \sum_{s_{k+1} \le x} (s_{k+1} - s_k)^\gamma \sim B(\gamma) x \; \; \mbox{ with some constant} \; \; B(\gamma) > 0 \] is true for $0 \le \gamma < 3.75$. This improves the previous best range $0 \le \gamma < 3.6875$.