FormalConjectures.ErdosProblems.«945»

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import FormalConjecturesUtil

Erdős Problem 945

References:

erdosproblems.com/945

[ErMi52] Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.


open Filter Real


namespace Erdos945


abbrev τ  := fun (n : ℕ) => n.divisors.card

Let $F(x)$ be the maximal $k$ such that there exist $n+1, \dots, n+k \le x$ with $τ(n+1), \dots, τ(n+k)$ all distinct, where $τ(m)$ counts the divisors of $m$.

noncomputable def F (x : ℝ) : ℕ :=
  sSup {k | ∃ (n : ℕ), n + k ≤ x ∧ (Set.Ioc n (n + k)).InjOn τ}


-- Implementation note: we define a Prop here and below to be able to easily formulate
-- the equivalence between the two variants. Because the theorems require `answer(sorry)` we
-- can't handle this with `type_of%`.
def Erdos945Prop : Prop := ∃ O : ℝ → ℝ, O =O[atTop] (1 : ℝ → ℝ) ∧ ∀ᶠ x in atTop, F x ≤ log x ^ O x

Is it true that $F(x) \leq (\log x)^{O(1)}$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_945 : answer(sorry) ↔ Erdos945Prop := by⊢ True ↔ Erdos945Prop
  sorryAll goals completed! 🐙


def Erdos945Constant : Prop :=
  ∃ C > (0 : ℝ),  ∀ᶠ x : ℝ in atTop,
    ∃ a b : ℕ, a ≠ b ∧
    ↑a ∈ Set.Icc x (x + log x ^ C) ∧
    ↑b ∈ Set.Icc x (x + log x ^ C) ∧
    τ a = τ b

Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_945.variants.constant : answer(sorry) ↔ Erdos945Constant := by⊢ True ↔ Erdos945Constant
  sorryAll goals completed! 🐙

The two ways of phrasing the conjecture are equivalent.

@[category textbook, AMS 11]
theorem declaration uses 'sorry'erdos_945.variants.equivalence : Erdos945Prop ↔ Erdos945Constant := by⊢ Erdos945Prop ↔ Erdos945Constant
  sorryAll goals completed! 🐙

Erdős and Mirsky [ErMi52] proved that $\frac{(\log x)^{1/2}}{\log\log x}\ll F(x)$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_945.variants.lower_bound :
    (fun (x : ℕ) => (log x).sqrt /(log x).log) =O[atTop] fun (n : ℕ) => (F n : ℝ) := by⊢ (fun x => √(log ↑x) / log (log ↑x)) =O[atTop] fun n => ↑(F ↑n)
  sorryAll goals completed! 🐙

Erdős and Mirsky [ErMi52] proved that $\log F(x) \ll \frac{(\log x)^{1/2}}$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_945.variants.upper_bound :
    (fun (n : ℕ) => (F n : ℝ).log) =O[atTop] fun (x : ℕ) => (log x).sqrt /(log x).log := by⊢ (fun n => log ↑(F ↑n)) =O[atTop] fun x => √(log ↑x) / log (log ↑x)
  sorryAll goals completed! 🐙


-- TODO(firsching): add observations what follows from Cramér's conjecture and if every sufficient
-- interval contains a squarefree number.

end Erdos945