Erdős problem 950 · problems.science

This function was considered by de Bruijn, Erdős, and Turán, who showed that $\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x$ . 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 statements2415f78e850a

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FormalConjectures/ErdosProblems/

950.lean

Retained formal statement7 of 7

He notes that if $\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)$ for all $y<x-(\log x)^C$ for some constant $C>0$ then $f(n)\ll \log\log\log n$ .

FormalConjectures/ErdosProblems/950.leanErdos950.erdos_950.variants.weaker_pi_implies_f5 linesExact file

1(∃ C > 0,2    ∃ K > 0,3      ∀ᶠ (x : ℕ) in Filter.atTop,4        ∀ (y : ℕ), ↑y < ↑x - Real.log ↑x ^ C → ↑x.primeCounting < ↑y.primeCounting + K * ((↑x - ↑y) / Real.log ↑x)) →5  Erdos950.f =O[Filter.atTop] fun n => Real.log (Real.log (Real.log ↑n))

SolvedStatement only, no proof

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