1. FormalConjectures
  2. ErdosProblems
  3. «538»
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Erdős Problem 538

Reference: erdosproblems.com/538

@[expose] public sectionopen Asymptotics Filternamespace Erdos538open scoped Classical in

The representations m = p * a with p prime and a ∈ A.

def representations (A : Finset ℕ) (m : ℕ) : Finset (ℕ × ℕ) := (Finset.range (m + 1) ×ˢ A).filter (fun pa => Nat.Prime pa.1 ∧ m = pa.1 * pa.2)

A ⊆ {1, …, N} and every m has at most r representations m = p a.

def Admissible (r N : ℕ) (A : Finset ℕ) : Prop := (∀ a ∈ A, 1 ≤ a ∧ a ≤ N) ∧ ∀ m : ℕ, (representations A m).card ≤ r

The reciprocal sum ∑_{n ∈ A} 1/n of the problem.

def reciprocalMass (A : Finset ℕ) : ℚ := ∑ a ∈ A, (1 : ℚ) / a

The largest reciprocal sum ∑_{n ∈ A} 1/n over admissible A ⊆ {1, …, N}.

noncomputable def maxMass (r N : ℕ) : ℝ := sSup ((fun A => (reciprocalMass A : ℝ)) '' {A : Finset ℕ | Admissible r N A})

Let $r\geq 2$ and suppose that $A\subseteq{1,\ldots,N}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$. Give the best possible upper bound for $\sum_{n\in A}\frac{1}{n}$.

Erdős observed that $\sum_{n\in A}\frac{1}{n}\ll r\frac{\log N}{\log\log N}$, and the order Θ_r(log N / loglog N) is known (see erdos_538.matching_order). The best possible upper bound is the asymptotic size of the largest reciprocal sum maxMass r N over admissible A.

@[category research open, AMS 11] theorem erdos_538 : let f : ℕ → ℕ → ℝ := answer(sorry) ∀ r : ℕ, 2 ≤ r → maxMass r ~[atTop] f r := ⊢ let f := sorry; ∀ (r : ℕ), 2 ≤ r → maxMass r ~[atTop] f r All goals completed! 🐙

The reciprocal sum has matching order Θ_r(log N / loglog N): an explicit upper bound for every admissible A, together with a witnessing construction achieving the same order. This pins the order (up to the one iterated-logarithm factor) but not the best possible upper bound asked for in erdos_538.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-538/Research/FinalMatchingOrder.lean"] theorem erdos_538.matching_order (r N : ℕ) (hr : 2 ≤ r) (hN : 2 ≤ N) : (∀ A : Finset ℕ, Admissible r N A → Real.log (Real.log (N + 1)) * (reciprocalMass A : ℝ) ≤ 2 * r * (1 + Real.log (N * N))) ∧ (∃ A : Finset ℕ, Admissible r N A ∧ Real.log (N + 1) ≤ 4 + (8192 * (Nat.log 2 (Nat.log 2 N) + 1) : ℝ) * (reciprocalMass A : ℝ)) := r:ℕN:ℕhr:2 ≤ rhN:2 ≤ N⊢ (∀ (A : Finset ℕ), Admissible r N A → Real.log (Real.log (↑N + 1)) * ↑(reciprocalMass A) ≤ 2 * ↑r * (1 + Real.log (↑N * ↑N))) ∧ ∃ A, Admissible r N A ∧ Real.log (↑N + 1) ≤ 4 + 8192 * (↑(Nat.log 2 (Nat.log 2 N)) + 1) * ↑(reciprocalMass A) All goals completed! 🐙end Erdos538