FormalConjectures.ErdosProblems.«462»

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import FormalConjecturesUtil

Erdős Problem 462

Reference: erdosproblems.com/462

namespace Erdos462open Filteropen scoped BigOperators

Let $p(n)$ denote the least prime factor of $n$. Is there a constant $C>0$ such that $$\sum_{x\leq n\leq x+C\sqrt{x}(\log x)^2}\frac{p(n)}{n}\gg 1$$ for all sufficiently large $x$?

@[category research open, AMS 11]
theorem erdos_462 : answer(sorry) ↔
    ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ᶠ x : ℕ in atTop,
      c ≤ ∑ n ∈ Finset.Icc x
        ⌊(x : ℝ) + C * Real.sqrt x * (Real.log x) ^ 2⌋₊, (n.minFac : ℝ) / n := by⊢ True ↔
  ∃ C c, 0 < C ∧ 0 < c ∧ ∀ᶠ (x : ℕ) in atTop, c ≤ ∑ n ∈ Finset.Icc x ⌊↑x + C * √↑x * Real.log ↑x ^ 2⌋₊, ↑n.minFac / ↑n
  sorryAll goals completed! 🐙

end Erdos462