Elliott–Halberstam conjecture

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import FormalConjecturesUtil

/-!
# Elliott–Halberstam conjecture

*References:*
- [Wikipedia](https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture)
- [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.
-/

namespace ElliottHalberstamConjecture

/--
$\pi(x; q, a)$: the number of primes $p \le x$ with $p \equiv a \pmod q$.
-/
def primesInAPCount (x q : ℕ) (a : ZMod q) : ℕ :=
  ((Finset.range (x + 1)).filter fun p : ℕ => p.Prime ∧ (p : ZMod q) = a).card

/--
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.
-/
noncomputable def E (x q : ℕ) : ℝ :=
  ⨆ a : (ZMod q)ˣ, |(primesInAPCount x q a : ℝ) - (x.primeCounting : ℝ) / (q.totient : ℝ)|

/--
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 $x > 2$.
-/
@[category research open, AMS 11]
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
  sorry

/--
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.
-/
@[category research solved, AMS 11]
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
  sorry

/--
The Elliott–Halberstam conjecture fails at the endpoint $\theta = 1$, as shown by
Friedlander and Granville [FG89].
-/
@[category research solved, AMS 11]
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
  sorry

end ElliottHalberstamConjecture

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