Erdős problem 145 · problems.science

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?

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FormalConjectures/ErdosProblems/

145.lean

Retained formal statement4 of 4

Erdős [Er51] proved this for all $0\leq \alpha\leq 2$ .

[Er51] Erdös, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.

FormalConjectures/ErdosProblems/145.leanErdos145.erdos_145.variants.le_two5 linesExact file

1∀ {α : ℝ},2  α ∈ Set.Icc 0 2 →3    ∃ β,4      Filter.Tendsto (fun x => 1 / x * ∑ n ∈ Erdos145.A x, (↑(Erdos145.s (n + 1)) - ↑(Erdos145.s n)) ^ α) Filter.atTop5        (nhds β)

SolvedStatement only, no proof

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