FormalConjectures/ErdosProblems/1060.lean

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L4DEL
Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1060

Reference: erdosproblems.com/1060

Imports (1)
  1. theorem Erdos1060.erdos_1060.parts.i The 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 $k$ sorry line 40 source

    Source documentation

    The 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 $\log n$.

    depends on sorry what this means

    Statement uses: Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, HDiv.hDiv, OfNat.ofNat, Real.log, Filter.Eventually, LE.le, Finset.card, Finset.filter, Eq, HMul.hMul, ArithmeticFunction, ArithmeticFunction.sigma, Finset.Iic, HPow.hPow

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    46 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    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 := by sorry
…
  2. theorem Erdos1060.erdos_1060.parts.ii Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$ sorry line 46 source

    Source documentation

    Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$.

    depends on sorry what this means

    Statement uses: Exists, Real, Asymptotics.IsBigO, Nat, Filter.atTop, Finset.card, Finset.filter, Eq, HMul.hMul, ArithmeticFunction, ArithmeticFunction.sigma, OfNat.ofNat, Finset.Iic, HPow.hPow, Real.log

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    34 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem erdos_1060.parts.ii :
    ∃ (C : ℝ), (fun n ↦ (#{k ≤ n | k * σ 1 k = n} : ℝ)) =O[atTop]
      (fun n ↦ log n ^ C) := by sorry
…