Positivity of the product of local densities

Proved

Vino.prod_threePrimeFactor_pos

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

analytic-number-theorycircle-methodnumber-theorysingular-series

For any finite set $s$ of primes and any odd $n$ ,

\prod_{p\in s}\mathfrak S_p(n)>0 .

Every partial Euler product of the three primes singular series is therefore strictly positive at odd $n$ .

Preamble

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

Formal statement

namespace Vino

theorem prod_threePrimeFactor_pos {s : Finset ℕ} (hs : ∀ p ∈ s, Nat.Prime p) {n : ℤ} (hn : ¬ (2 : ℤ) ∣ n) :
    0 < ∏ p ∈ s, threePrimeFactor p n := 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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