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Erdős problem 950

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This function was considered by de Bruijn, Erdős, and Turán, who showed that ∑n<xf(n)∼∑n<xf(n)2∼x\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x. They gave no proofs, but a proof of the (harder) second claim is given by Gorodetsky here [mathoverflow/508491].

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7 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

950.lean

Retained formal statement6 of 7

Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every ϵ>0\epsilon>0, if xx is sufficiently large there exists y<xy<x such that π(x)<π(y)+ϵπ(x−y)\pi(x)< \pi(y)+\epsilon \pi(x-y). Compare this to [855].

FormalConjectures/ErdosProblems/950.leanErdos950.erdos_950.variants.weaker_pi1 lineExact file
True ↔ ∀ ε > 0, ∀ᶠ (x : ℕ) in Filter.atTop, ∃ y < x, ↑x.primeCounting < ↑y.primeCounting + ε * ↑(x - y).primeCounting
OpenStatement only, no proof