The three primes singular series is nonzero at odd $n$

Proved

Vino.singSeriesDvd_ne_zero_of_odd

by tabbott · Sep 2, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theorycircle-methodnumber-theorysingular-series

Let $Q$ be squarefree and let $n$ be odd. Then

\sum_{q\mid Q}\frac{\mu(q)c_q(n)}{\varphi(q)^3}\neq0 .

Every truncation of the three primes singular series at a squarefree modulus is nonzero — in fact positive — for odd $n$ . This is the complement of the vanishing at even $n$ , and it is the local input to the assertion that the main term of the three primes asymptotic does not degenerate.

Preamble

import Definitions.Def_Vino_ramanujan
import Mathlib.Data.Nat.Totient
import Mathlib.Algebra.BigOperators.Ring.Finset
open Finset

Formal statement

namespace Vino

theorem singSeriesDvd_ne_zero_of_odd {Q : ℕ} (hQ : Squarefree Q) {n : ℤ} (hn : ¬ (2 : ℤ) ∣ n) :
    singSeriesDvd Q n ≠ 0 := by sorry

end Vino

Source

R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997, Chapter 3, Section 3.2 (the singular series of the three primes theorem); I. M. Vinogradov, Representation of an odd number as a sum of three primes, Doklady Akademii Nauk SSSR 15 (1937), 291-294.

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The three primes singular series is nonzero at odd nnn

The three primes singular series is nonzero at odd $n$