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import FormalConjecturesUtilErdős Problem 942
open Nat Filter Topology
namespace Erdos942
Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$.
def erdos_942.h (n : ℕ) : ℕ := ((Finset.Ico (n ^ 2) ((n + 1) ^ 2)).filter Powerful).card
Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$.
@[category research open, AMS 11]
theorem erdos_942 : answer(sorry) ↔ ∃ c > 0, ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧
(∀ᶠ n in atTop, erdos_942.h n < (Real.log n) ^ (c + o n)) ∧
{n | erdos_942.h n > (Real.log n) ^ (c - o n)}.Infinite := ⊢ True ↔
∃ c > 0,
∃ o,
o =o[atTop] 1 ∧
(∀ᶠ (n : ℕ) in atTop, ↑(erdos_942.h n) < Real.log ↑n ^ (c + o n)) ∧
{n | ↑(erdos_942.h n) > Real.log ↑n ^ (c - o n)}.Infinite
All goals completed! 🐙
It is not hard to prove that $\limsup h(n) = \infty$.
@[category textbook, AMS 11]
theorem erdos_942.variants.limsup :
atTop.limsup (((fun (n : ℕ) ↦ (n : ℕ∞)) ∘ erdos_942.h)) = ⊤ := ⊢ Filter.limsup ((fun n => ↑n) ∘ h) atTop = ⊤
All goals completed! 🐙
It is not hard to prove that the density $\delta_l$ of integers for which $h(n) = l$ exists and satisfies $$\sum_l \delta_l = 1$$.
@[category textbook, AMS 11]
theorem erdos_942.variants.density :
∃ δ : ℕ → ℝ, ∀ l, {n | erdos_942.h n = l}.HasDensity (δ l) ∧
∑' l, δ l = 1 := ⊢ ∃ δ, ∀ (l : ℕ), {n | h n = l}.HasDensity (δ l) ∧ ∑' (l : ℕ), δ l = 1
All goals completed! 🐙
end Erdos942