FormalConjectures.ErdosProblems.«417»

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import FormalConjecturesUtil

Erdős Problem 417

References:

erdosproblems.com/417

[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.


open Nat Set Filter

open scoped Topology


namespace Erdos417

Let$$V'(x)=#{\phi(m) : 1\leq m\leq x}$$and$$V(x)=#{\phi(m) \leq x : 1\leq m}.$$ Does $\lim V(x)/V'(x)$ exist?

Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x) to ensure the limit is finite (bounded between 0 and 1).

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_417.parts.i :
    answer(sorry) ↔ ∃ L : ℝ, Tendsto (fun x ↦
      ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ℝ) /
      ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ℝ))
      atTop (𝓝 L) := by⊢ True ↔ ∃ L, Tendsto (fun x => ↑(φ '' {m | 1 ≤ m ∧ ↑m ≤ x}).ncard / ↑{k | k ∈ range φ ∧ ↑k ≤ x}.ncard) atTop (𝓝 L)
  sorryAll goals completed! 🐙

Is it $>1$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_417.parts.ii :
    answer(sorry) ↔ ∃ L < 1, Tendsto (fun x ↦
      ((totient '' { m | 1 ≤ m ∧ (m : ℝ) ≤ x }).ncard : ℝ) /
      ({ k | k ∈ range totient ∧ (k : ℝ) ≤ x }.ncard : ℝ))
      atTop (𝓝 L) := by⊢ True ↔ ∃ L < 1, Tendsto (fun x => ↑(φ '' {m | 1 ≤ m ∧ ↑m ≤ x}).ncard / ↑{k | k ∈ range φ ∧ ↑k ≤ x}.ncard) atTop (𝓝 L)
  sorryAll goals completed! 🐙


end Erdos417