FormalConjectures.ErdosProblems.«1057»

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import FormalConjecturesUtil

Erdős Problem 1057

References:

erdosproblems.com/1057

[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many Carmichael numbers. Ann. of Math. (2) (1994), 703--722.

[Er56c] Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956), 201--206.

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

[Ha08] Harman, Glyn, Watt's mean value theorem and Carmichael numbers. Int. J. Number Theory (2008), 241--248.

[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).

[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.


open Nat Real Filter Set

open scoped Topology Asymptotics


namespace Erdos1057

Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$.

noncomputable def carmichaelCounting (x : ℝ) : ℝ :=
  ({n : ℕ | IsCarmichael n ∧ (n : ℝ) ≤ x}.ncard : ℝ)

Is it true that $C(x)=x^{1-o(1)}$?

This is discussed in problem A13 of Guy's collection [Gu04].

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1057 :
    answer(sorry) ↔ Tendsto (fun x ↦ Real.log (carmichaelCounting x) / Real.log x) atTop (𝓝 1) := by⊢ True ↔ Tendsto (fun x => Real.log (carmichaelCounting x) / Real.log x) atTop (𝓝 1)
  sorryAll goals completed! 🐙

Erdős [Er56c] proved $C(x) < x \exp\left(-c \frac{\log x\log\log\log x}{\log\log x}\right)$ for some constant $c>0$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.upper_bound :
    ∃ c > 0, ∀ᶠ x in atTop,
      carmichaelCounting x < x * Real.exp (-c * (Real.log x * Real.log (Real.log (Real.log x))) /
        Real.log (Real.log x)) := by⊢ ∃ c > 0,
  ∀ᶠ (x : ℝ) in atTop,
    carmichaelCounting x < x * rexp (-c * (Real.log x * Real.log (Real.log (Real.log x))) / Real.log (Real.log x))
  sorryAll goals completed! 🐙

Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.pomerance :
    answer(sorry) ↔ Tendsto (fun x ↦
      -(Real.log (carmichaelCounting x / x) * Real.log (Real.log x)) /
      (Real.log x * Real.log (Real.log (Real.log x)))) atTop (𝓝 1) := by⊢ True ↔
  Tendsto
    (fun x =>
      -(Real.log (carmichaelCounting x / x) * Real.log (Real.log x)) / (Real.log x * Real.log (Real.log (Real.log x))))
    atTop (𝓝 1)
  sorryAll goals completed! 🐙

Alford, Granville, and Pomerance [AGP94] proved that $C(x)\to \infty$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.agp_infinite :
    Tendsto carmichaelCounting atTop atTop := by⊢ Tendsto carmichaelCounting atTop atTop
  sorryAll goals completed! 🐙

Alford, Granville, and Pomerance [AGP94] proved that $C(x)>x^{2/7}$ for large $x$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.agp_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (2 / 7 : ℝ) := by⊢ ∀ᶠ (x : ℝ) in atTop, carmichaelCounting x > x ^ (2 / 7)
  sorryAll goals completed! 🐙

The lower bound $C(x)> x^{0.33336704}$ was proved by Harman [Ha08].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.harman_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (0.33336704 : ℝ) := by⊢ ∀ᶠ (x : ℝ) in atTop, carmichaelCounting x > x ^ 0.33336704
  sorryAll goals completed! 🐙

This exponent was improved to $0.3389$ by Lichtman [Li22].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_1057.variants.lichtman_lower_bound :
    ∀ᶠ x in atTop, carmichaelCounting x > x ^ (0.3389 : ℝ) := by⊢ ∀ᶠ (x : ℝ) in atTop, carmichaelCounting x > x ^ 0.3389
  sorryAll goals completed! 🐙


end Erdos1057