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import FormalConjecturesUtilErdős Problem 1060
open Asymptotics Finset Filter Realopen scoped ArithmeticFunction.sigma
namespace Erdos1060The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of $\log n$.
@[category research open, AMS 11]
theorem erdos_1060.parts.i :
∃ h : ℕ → ℝ,
h =o[atTop] (fun n ↦ 1 / log (log n)) ∧ ∀ᶠ n in atTop, #{k ≤ n | k * σ 1 k = n} ≤ (n : ℝ) ^ h n := ⊢ ∃ h, (h =o[atTop] fun n => 1 / log (log ↑n)) ∧ ∀ᶠ (n : ℕ) in atTop, ↑(#({k ∈ Iic n | k * (σ 1) k = n})) ≤ ↑n ^ h n All goals completed! 🐙Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$.
@[category research open, AMS 11]
theorem erdos_1060.parts.ii :
∃ (C : ℝ), (fun n ↦ (#{k ≤ n | k * σ 1 k = n} : ℝ)) =O[atTop]
(fun n ↦ log n ^ C) := ⊢ ∃ C, (fun n => ↑(#({k ∈ Iic n | k * (σ 1) k = n}))) =O[atTop] fun n => log ↑n ^ C All goals completed! 🐙
end Erdos1060