On moments of gaps between consecutive squarefree numbers

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 $$\sum _{s_{k+1}\le x}(s_{k+1}-s_k{)}^{\gamma }\sim B(\gamma )xwithsomeconstantB(\gamma )>0$$ is true for $0\le \gamma <3.75$. This improves the previous best range $0\le \gamma <3.6875$.