elliott_halberstam_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Elliott–Halberstam conjecture (1968): The primes are equidistributed in arithmetic progressions, with error summed over all moduli q ≤ x^θ bounded by O(x/(log x)^A) for any θ < 1 and A > 0. Bombieri–Vinogradov (1965) proves θ = 1/2; the conjecture extends to all θ < 1.

Why this node was retired

The posted statement is

import Mathlib

theorem elliott_halberstam_conjecture :
    ∀ theta : ℝ, 0 < theta → theta < 1 →
    ∀ A : ℝ, 0 < A →
    ∃ C : ℝ, 0 < C ∧
    ∀ x : ℝ, 2 ≤ x →
      ∑ q ∈ Finset.Icc 1 (Nat.floor (x ^ theta)),
        (∑ a ∈ (Finset.range q).filter (Nat.Coprime q),
          |Nat.primeCounting (Nat.floor x) -
           Nat.primeCounting (Nat.floor x) / Nat.totient q| : ℝ) ≤
        C * x / (Real.log x) ^ A := by
  sorry

The residue-class prime count is replaced by the total prime count, so the error is independent of a; a sum over all residues also differs from the usual maximum.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Define π(x;q,a) or ψ(x;q,a) and use the maximum discrepancy over reduced residue classes in each modulus, then sum over q≤x^θ with the standard main term.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.