The local factor at $2$ equals $2$ for odd $n$

Proved

Vino.threePrimeFactor_two_of_odd

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

analytic-number-theorycircle-methodnumber-theorysingular-series

For odd $n$ the local factor of the three primes singular series at $p=2$ is

\mathfrak S_2(n)=1+\frac{1}{(2-1)^3}=2 .

The factor $2$ is the density gain from the fact that every prime except $2$ is odd, so the three summands are forced into the odd residue class, which is half of all residues but carries all the primes.

Preamble

import Definitions.Def_Vino_ramanujan
import Mathlib.Data.Nat.Totient
open Finset

Formal statement

namespace Vino

theorem threePrimeFactor_two_of_odd {n : ℤ} (h : ¬ (2 : ℤ) ∣ n) : threePrimeFactor 2 n = 2 := 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.

View graph

The local factor at 222 equals 222 for odd nnn

The local factor at $2$ equals $2$ for odd $n$