The density of squarefree numbers is well known.
I am wondering about the squarefree numbers in $[2,n]$ namely the second and higher moments of the gaps. Is anything known about these?
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3$\begingroup$ Since there are asymptotically $\frac{6}{\pi^2}n$ squarefree numbers in the interval, the average gap is going to have length $\frac{\pi^2}{6}$. The second moment is definitely a more interesting question. $\endgroup$– WojowuCommented Jan 31, 2018 at 21:16
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1 Answer
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A number of authors e.g. Hooley, Filaseta, Trifonov have considered this problem of moments of gaps between square-free numbers. For example, Filaseta and Trifonov (paper in Proc. London Math. Soc. (1996); see Theorem 4) showed that for all $0\le \gamma < 43/13=3.30769\ldots$ one has $$ \sum_{s_{n+1}\le x} (s_{n+1}-s_n)^{\gamma} \sim B(\gamma) x, $$ for some constant $B(\gamma)$. Here $s_n$ denotes the $n$-th square-free number.
