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 Filteropen scoped Topologynamespace 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?

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

Is it $>1$?

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

end Erdos417