ElliottHalberstamConjecture.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Elliott–Halberstam conjecture

References:

Wikipedia
[EH68] Elliott, Peter D. T. A. and Halberstam, Heini, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), 59–72.
[FG89] Friedlander, John and Granville, Andrew, Limitations to the equi-distribution of primes I, Ann. of Math. (2) 129 (1989), no. 2, 363–382.

Imports (1)

FormalConjecturesUtil

def ElliottHalberstamConjecture.primesInAPCount $\pi(x; q, a)$: the number of primes $p \le x$ with $p \equiv a \pmod q$ line 38 source
Source documentation
$\pi(x; q, a)$: the number of primes $p \le x$ with $p \equiv a \pmod q$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ZMod
Proof uses: Finset.card, Finset.filter, And, Nat.Prime, Eq, Finset.range, HAdd.hAdd, OfNat.ofNat
12 instances, coercions and auxiliary constants left out.
```
def primesInAPCount (x q : ℕ) (a : ZMod q) : ℕ :=
…
```
def ElliottHalberstamConjecture.E The error term line 47 source
Source documentation
The error term $$E(x; q) = \max_{\gcd(a,q)=1} \left|\pi(x;q,a) - \frac{\pi(x)}{\varphi(q)}\right|,$$ measuring the deviation of the primes in the arithmetic progressions modulo $q$ from uniform distribution among the $\varphi(q)$ coprime residue classes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: iSup, Units, ZMod, abs, HSub.hSub, ElliottHalberstamConjecture.primesInAPCount, HDiv.hDiv, Nat.primeCounting, Nat.totient
15 instances, coercions and auxiliary constants left out.
```
noncomputable def E (x q : ℕ) : ℝ :=
…
```
theorem ElliottHalberstamConjecture.elliott_halberstam Documented declaration: elliott_halberstam sorry line 57 source
Source documentation
The Elliott–Halberstam conjecture: for every $\theta < 1$ and $A > 0$ there exists a constant $C > 0$ such that $$\sum_{1 \le q \le x^{\theta}} E(x; q) \le \frac{C x}{\log^A x}$$ for all $x > 2$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, And, GT.gt, Nat, LE.le, Finset.sum, Finset.Icc, Nat.floor, HPow.hPow, ElliottHalberstamConjecture.E, HDiv.hDiv, HMul.hMul, 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 elliott_halberstam (θ : ℝ) (hθ : θ < 1) (A : ℝ) (hA : 0 < A) :
    ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →
      ∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A := by
…
```
Mentions: ElliottHalberstamConjecture.E
theorem ElliottHalberstamConjecture.elliott_halberstam.variants.bombieri_vinogradov The Bombieri–Vinogradov theorem: the Elliott–Halberstam conjecture holds for every $\theta < 1/2$ sorry line 68 source
Source documentation
The Bombieri–Vinogradov theorem: the Elliott–Halberstam conjecture holds for every $\theta < 1/2$. This result may be regarded as an averaged form of the generalized Riemann hypothesis.

depends on sorry what this means
Statement uses: Real, LT.lt, HDiv.hDiv, OfNat.ofNat, Nat, Exists, And, GT.gt, LE.le, Finset.sum, Finset.Icc, Nat.floor, HPow.hPow, ElliottHalberstamConjecture.E, HMul.hMul, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem elliott_halberstam.variants.bombieri_vinogradov (θ : ℝ) (hθ : θ < 1 / 2) (A : ℝ)
    (hA : 0 < A) :
    ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →
      ∑ q ∈ Finset.Icc 1 ⌊(x : ℝ) ^ θ⌋₊, E x q ≤ C * x / Real.log x ^ A := by
…
```
Mentions: ElliottHalberstamConjecture.E
theorem ElliottHalberstamConjecture.elliott_halberstam.variants.friedlander_granville The Elliott–Halberstam conjecture fails at the endpoint $\theta = 1$, as shown by Friedlander and Granville [FG89] sorry line 79 source
Source documentation
The Elliott–Halberstam conjecture fails at the endpoint $\theta = 1$, as shown by Friedlander and Granville [FG89].

depends on sorry what this means
Statement uses: Not, Real, LT.lt, OfNat.ofNat, Exists, And, GT.gt, Nat, LE.le, Finset.sum, Finset.Icc, ElliottHalberstamConjecture.E, HDiv.hDiv, HMul.hMul, HPow.hPow, 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 elliott_halberstam.variants.friedlander_granville :
    ¬ ∀ A : ℝ, 0 < A → ∃ C > (0 : ℝ), ∀ x : ℕ, 2 < x →
      ∑ q ∈ Finset.Icc 1 x, E x q ≤ C * x / Real.log x ^ A := by
…
```
Mentions: ElliottHalberstamConjecture.E