primes_ap_quantitative

⚠️ 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.

Quantitative Dirichlet theorem: The primes in arithmetic progressions a (mod q) have density 1/φ(q) among all primes. Proved by Dirichlet (1837) qualitatively. Quantitative form with explicit error terms O(x exp(-c√log x)) by Siegel-Walfisz.

Why this node was retired

The posted statement is

import Mathlib

theorem primes_ap_quantitative (q a : ℕ) (hq : 2 ≤ q) (ha : Nat.Coprime q a) :
    Filter.Tendsto (fun x : ℝ =>
      (∑ p ∈ (Finset.range (Nat.floor x)).filter (fun n => Nat.Prime n ∧ n % q = a),
        (1 : ℝ)) /
      (x / (Real.log x * Nat.totient q)))
    Filter.atTop (nhds 1) := by
  sorry

The counted residue condition compares a remainder to an unrestricted representative.

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

Proposed corrected statement

Replace p%q=a with p%q=a%q, or add a<q.

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.