FormalConjectures.ErdosProblems.«912»

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import FormalConjecturesUtil

Erdős Problem 912

References:

erdosproblems.com/912

[Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.


open scoped Nat Asymptotics

open Filter


namespace Erdos912

If $n! = \prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$ to be the number of distinct exponents $k_i$.

noncomputable def h (n : ℕ) : ℕ := (n !).factorization.frange.card

Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_912.variants.selfridge :
    (fun n => (h n : ℝ)) =Θ[atTop] (fun n => (n / Real.log n) ^ (1 / 2 : ℝ)) := by⊢ (fun n => ↑(h n)) =Θ[atTop] fun n => (↑n / Real.log ↑n) ^ (1 / 2)
  sorryAll goals completed! 🐙

Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_912 : ∃ c > 0,
    (fun n => (h n : ℝ)) ~[atTop] (fun n => c * (n / Real.log n) ^ (1 / 2 : ℝ)) := by⊢ ∃ c > 0, (fun n => ↑(h n)) ~[atTop] fun n => c * (↑n / Real.log ↑n) ^ (1 / 2)
  sorryAll goals completed! 🐙

A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_912.variants.tao :
    (fun n => (h n : ℝ)) ~[atTop] (fun n => √(2 * Real.pi) * (n / Real.log n) ^ (1 / 2 : ℝ)) := by⊢ (fun n => ↑(h n)) ~[atTop] fun n => √(2 * Real.pi) * (↑n / Real.log ↑n) ^ (1 / 2)
  sorryAll goals completed! 🐙


end Erdos912