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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.

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 declaration uses 'sorry'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 := θ:ℝhθ:θ < 1A:ℝhA:0 < A⊢ ∃ C > 0, ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 ⌊↑x ^ θ⌋₊, E x q ≤ C * ↑x / Real.log ↑x ^ A All goals completed! 🐙

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 declaration uses 'sorry'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 := θ:ℝhθ:θ < 1 / 2A:ℝhA:0 < A⊢ ∃ C > 0, ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 ⌊↑x ^ θ⌋₊, E x q ≤ C * ↑x / Real.log ↑x ^ A All goals completed! 🐙

The Elliott–Halberstam conjecture fails at the endpoint $\theta = 1$, as shown by Friedlander and Granville [FG89].

@[category research solved, AMS 11] theorem declaration uses 'sorry'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 := ⊢ ¬∀ (A : ℝ), 0 < A → ∃ C > 0, ∀ (x : ℕ), 2 < x → ∑ q ∈ Finset.Icc 1 x, E x q ≤ C * ↑x / Real.log ↑x ^ A All goals completed! 🐙 end ElliottHalberstamConjecture