726.lean`

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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 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.

Imports (1)

FormalConjecturesUtil

theorem Erdos726.erdos_726 As $n\to \infty$ ranges over integers $\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}$? sorry line 46 source
Source documentation
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 $p/2<r<p$. The remainder n % p is computed in ℕ before casting to ℝ.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Finset.sum, Finset.filter, And, Nat.Prime, LT.lt, HDiv.hDiv, OfNat.ofNat, HMod.hMod, Finset.range, HAdd.hAdd, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_726 :
    answer(sorry) ↔
      (fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter
          (fun p : ℕ ↦ p.Prime ∧ (p : ℝ) / 2 < ((n % p : ℕ) : ℝ)),
        (1 : ℝ) / (p : ℝ))
      ~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ)) / 2) := by
…
```
theorem Erdos726.erdos_726.variants.mertens_estimate The classical estimate of Mertens states that $\sum_{p\leq n}\frac{1}{p}\sim \log\log n$ sorry line 58 source
Source documentation
The classical estimate of Mertens states that $\sum_{p\leq n}\frac{1}{p}\sim \log\log n$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Finset.sum, Finset.filter, Nat.Prime, Finset.range, HAdd.hAdd, OfNat.ofNat, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem 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
…
```