Erdős problem 538 · problems.science

If each integer has at most $r$ representations $m = pa$ with $p$ prime and $a \in A \subseteq [1, N]$ , what is the best upper bound for $\sum_{a \in A} 1/a$ ? The candidate proof gives the matching order $\Theta_r(\log N / \log\log N)$ .

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FormalConjectures/ErdosProblems/

538.lean

Retained formal statement1 of 2

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 $\sum_{n\in A}\frac{1}{n}$ .

The order is known — ∑ 1/n = Θ_r(log N / loglog N) (see erdos_538.matching_order) — but the sharp constant is not. This asks whether maxMass r N has a well-defined leading constant c_r in c_r · log N / loglog N.

FormalConjectures/ErdosProblems/538.leanErdos538.erdos_5386 linesExact file

1True ↔2  ∀ (r : ℕ),3    2 ≤ r →4      ∃ c,5        0 < c ∧6          Filter.Tendsto (fun N => Erdos538.maxMass r N * Real.log (Real.log ↑N) / Real.log ↑N) Filter.atTop (nhds c)

OpenStatement only, no proof

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