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Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$, \[\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha\] exists?
#145
:
[Er79]
number theory
Erdős
[Er51]
proved this for all $0\leq \alpha \leq 2$, and Hooley
[Ho73]
extended this to all $\alpha \leq 3$.
See also
[208]
.
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