FormalConjectures.ErdosProblems.«726»

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Erdős Problem 726

References:

erdosproblems.com/726

[EGRS75] Erdős, P., and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\sp{2n}\sb{n})$. Math. Comp. (1975), 83-92.


open Nat Filter Finset

open scoped Topology Asymptotics


namespace Erdos726

As $n\to \infty$ ranges over integers $\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}$?

A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75].

By $n\in (p/2,p)\pmod{p}$ we mean $n\equiv r\pmod{p}$ for some integer $r$ with $p/2<r<p$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_726 :
    answer(sorry) ↔
      (fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter
          (fun p : ℕ ↦ p.Prime ∧ (p : ℝ) / 2 < (n % p : ℝ)),
        (1 : ℝ) / (p : ℝ))
      ~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ)) / 2) := by⊢ True ↔
  (fun n => ∑ p ∈ range (n + 1) with Nat.Prime p ∧ ↑p / 2 < ↑n % ↑p, 1 / ↑p) ~[atTop] fun n =>
    Real.log (Real.log ↑n) / 2
  sorryAll goals completed! 🐙

The classical estimate of Mertens states that $\sum_{p\leq n}\frac{1}{p}\sim \log\log n$.

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_726.variants.mertens_estimate :
    (fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter (fun p ↦ p.Prime), (1 : ℝ) / ((p : ℕ) : ℝ))
    ~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ))) := by⊢ (fun n => ∑ p ∈ range (n + 1) with Nat.Prime p, 1 / ↑p) ~[atTop] fun n => Real.log (Real.log ↑n)
  sorryAll goals completed! 🐙


end Erdos726