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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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

145.lean

Retained formal statement2 of 4

Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \leq \alpha\leq 11/3$ .

[GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and their Applications in Number Theory. (1997).

FormalConjectures/ErdosProblems/145.leanErdos145.erdos_145.variants.le_eleven_thirds5 linesExact file

1∀ {α : ℝ},2  α ∈ Set.Icc 0 (11 / 3) →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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