Erdős problem 377 · problems.science

Is there some absolute constant $C > 0$ such that $\sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C$ for all $n$ ?

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FormalConjectures/ErdosProblems/

377.lean

Retained formal statement4 of 5

Erdos, Graham, Ruzsa, and Straus proved that if $f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p}$ and $\gamma_0 = \sum_{k = 2}^{\infty} \frac{\log k}{2^k}$ then $\lim_{x\to\infty} \frac{1}{x}\sum_{n\leq x} f(n)^2 = \gamma_0^2$

[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\binom{2n}{n}$ _. Math. Comp. (1975), 83-92.

FormalConjectures/ErdosProblems/377.leanErdos377.erdos_377.variants.limit.ii4 linesExact file

1∀ (γ₀ : ℝ),2  γ₀ = ∑' (k : ℕ), Real.log (↑k + 2) / 2 ^ (k + 2) →3    Filter.Tendsto (fun x => 1 / ↑x * ∑ n ∈ Finset.Icc 1 x, Erdos377.sumInvPrimesNotDvdCentralBinom n ^ 2) Filter.atTop4      (nhds (γ₀ ^ 2))

SolvedStatement only, no proof

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