Trivial bound for the twisted von Mangoldt sum

Proved

Vino.norm_vmSumChar_le

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

analytic-number-theorycircle-methoddirichlet-charactersnumber-theory

For every Dirichlet character $\chi$ modulo $q$ ,

|\psi(N,\chi)|=\left|\sum_{n<N}\Lambda(n)\chi(n)\right|\;\le\;\sum_{n<N}\Lambda(n)=\psi(N).

The trivial bound, attained by the principal character. The whole content of the prime number theorem in arithmetic progressions is that for non-principal $\chi$ this can be improved to $o(\psi(N))$ , and that improvement is what converts the major-arc identity into an asymptotic.

Preamble

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

Formal statement

namespace Vino

theorem norm_vmSumChar_le (q : ℕ) (χ : DirichletCharacter ℂ q) (N : ℕ) : ‖vmSumChar q χ N‖ ≤ psi N := 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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