FormalConjectures.ErdosProblems.«491»

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module

public import FormalConjecturesUtil

Erdős Problem 491

References:

erdosproblems.com/491

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.

[Wi70] E. Wirsing, A characterization of $\log n$ as an additive arithmetic function. Symposia Math. (1970), 45-57.

@[expose] public sectionopen Filternamespace Erdos491

Let $f : \mathbb{N} \to \mathbb{R}$ be an additive function (so that $f(ab) = f(a) + f(b)$ whenever $(a, b) = 1$). If $|f(n+1) - f(n)| < c$ for some constant $c$ and all $n$, then must there exist some $c'$ such that $f(n) = c' \log n + O(1)$?

A question of Erdős [Er61, p.237; Er82e, p.65], who had proved that $f(n) = c' \log n$ under the stronger hypotheses $f(n+1) - f(n) = o(1)$ or $f(n+1) \ge f(n)$. The answer is yes, proved by Wirsing [Wi70]. See also erdos_897.variants.log_growth.

@[category research solved, AMS 11, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos491.lean#L34"]
theorem erdos_491 : answer(True) ↔
    ∀ (f : ℕ → ℝ), (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
      (∃ C : ℝ, ∀ n : ℕ, |f (n + 1) - f n| < C) →
      ∃ c : ℝ, (fun n : ℕ ↦ f n - c * Real.log n) =O[atTop] (fun _ : ℕ ↦ (1 : ℝ)) := by⊢ True ↔
  ∀ (f : ℕ → ℝ),
    (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →
      (∃ C, ∀ (n : ℕ), |f (n + 1) - f n| < C) → ∃ c, (fun n ↦ f n - c * Real.log ↑n) =O[atTop] fun x ↦ 1
  sorryAll goals completed! 🐙

end Erdos491