233.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 233

References:

Imports (1)

FormalConjecturesUtil

theorem Erdos233.erdos_233 A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$ sorry line 40 source
Source documentation
A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Finset.sum, Finset.range, HPow.hPow, primeGap, OfNat.ofNat, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_233 :
    (fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) =O[atTop] fun N => N * (log N)^2 := by
…
```
Mentions: primeGap
theorem Erdos233.erdos_233.variants.upper_bound Cramér proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis sorry line 48 source
Source documentation
Cramér proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis.

depends on sorry what this means
Statement uses: RiemannHypothesis, Asymptotics.IsBigO, Nat, Real, Filter.atTop, Finset.sum, Finset.range, HPow.hPow, primeGap, OfNat.ofNat, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_233.variants.upper_bound (h : RiemannHypothesis) :
    (fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) =O[atTop] fun N => N * (log N)^4 := by
…
```
Mentions: primeGap
theorem Erdos233.erdos_233.variants.lower_bound The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$ for the sum of squares of gaps between consecutive primes sorry line 59 source
Source documentation
The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$ for the sum of squares of gaps between consecutive primes.
Formal proof linked here provided by AlphaProof.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HMul.hMul, HPow.hPow, Real.log, OfNat.ofNat, Finset.sum, Finset.range, primeGap
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_233.variants.lower_bound :
    (fun (N : ℕ) => N * (log N)^2) =O[atTop]
    (fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) := by
…
```
Mentions: primeGap