FormalConjectures.ErdosProblems.«821»

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import FormalConjecturesUtil

Erdős Problem 821

References:

erdosproblems.com/821

[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.

[Er35b] Erdős, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\varphi$-function. Quart. J. Math. (1935), 205-213.

[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.

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

[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Théor. Nombres Bordeaux (2011), 697--714.


open Nat Filter


namespace Erdos821

Let $g(n)$ count the number of $m$ such that $\phi(m)=n$.

noncomputable def g (n : ℕ) : ℕ :=
  { m : ℕ | totient m = n }.ncard

Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_821 :
    answer(sorry) ↔ ∀ ε > (0 : ℝ), { n : ℕ | (g n : ℝ) > (n : ℝ) ^ (1 - ε) }.Infinite := by⊢ True ↔ ∀ ε > 0, {n | ↑(g n) > ↑n ^ (1 - ε)}.Infinite
  sorryAll goals completed! 🐙

Pillai proved that $\limsup g(n)=\infty$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_821.variants.pillai :
    atTop.limsup (fun n : ℕ ↦ (g n : EReal)) = ⊤ := by⊢ limsup (fun n => ↑(g n)) atTop = ⊤
  sorryAll goals completed! 🐙

Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) > n^c$ for infinitely many $n$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_821.variants.erdos :
    ∃ c > (0 : ℝ), { n : ℕ | (n : ℝ) ^ c < (g n : ℝ) }.Infinite := by⊢ ∃ c > 0, {n | ↑n ^ c < ↑(g n)}.Infinite
  sorryAll goals completed! 🐙

The best known bound is that there are infinitely many $n$ such that $g(n) > n^{0.71568\cdots}$, obtained by Lichtman [Li22] as a consequence of proving that there are $\geq \frac{x}{(\log x)^{O(1)}}$ many primes $p\leq x$ such that all prime factors of $p-1$ are $\leq x^{0.2843\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman [BaHa98]).

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_821.variants.lichtman :
    ∃ c > (0.71568 : ℝ), { n : ℕ | (n : ℝ) ^ c < (g n : ℝ) }.Infinite := by⊢ ∃ c > 0.71568, {n | ↑n ^ c < ↑(g n)}.Infinite
  sorryAll goals completed! 🐙


end Erdos821