Empty character-twisted von Mangoldt sum

Proved

Vino.vmSumChar_zero_len

by tabbott · Sep 4, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theorycircle-methoddirichlet-charactersnumber-theory

The twisted sum $\psi(N,\chi)=\sum_{n<N}\Lambda(n)\chi(n)$ vanishes for $N=0$ :

\psi(0,\chi)=0.

Base case for induction on the length.

Preamble

import Definitions.Def_Vino_dirichlet
import Definitions.Def_Vino_primes
import Mathlib.NumberTheory.DirichletCharacter.Orthogonality
open Finset

Formal statement

namespace Vino

theorem vmSumChar_zero_len (q : ℕ) (χ : DirichletCharacter ℂ q) : vmSumChar q χ 0 = 0 := by sorry

end Vino

Source

H. Davenport, Multiplicative Number Theory, 3rd ed., Springer GTM 74, 2000, Chapters 9 and 26; R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997, Chapter 3, Section 3.1.

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