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Erdős problem 145

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Let s1<s2<⋯s_1 < s_2 < \cdots be the sequence of squarefree numbers. Is it true that, for any α≥0\alpha\geq 0, lim⁡x→∞1x∑sn≤x(sn+1−sn)α \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha exists?

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FormalConjectures/ErdosProblems/

145.lean

Retained formal statement3 of 4

Hooley [Ho73] extended this to all 0≤α≤30 \leq \alpha\leq 3.

[Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973.

FormalConjectures/ErdosProblems/145.leanErdos145.erdos_145.variants.le_three5 linesExact file
∀ {α : ℝ},  α ∈ Set.Icc 0 3 →    ∃ β,      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop        (nhds β)
SolvedStatement only, no proof