erdos_145

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?

theorem erdos_145 :
    answer(sorry) ↔ ∀ α ≥ (0 : ℝ), ∃ β : ℝ,
      atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β)

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