Trivial bound $|S(\alpha,N)|\le\psi(N)$

Proved

Vino.norm_vmSum_le

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

analytic-number-theorycircle-methodnumber-theoryprime-numbers

For every real $\alpha$ and every $N$ ,

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

This is the trivial bound: it is attained at $\alpha=0$ , and the entire minor-arc analysis consists of beating it by a power of $\log N$ or better.

Preamble

import Definitions.Def_Vino_primes
import Mathlib.Analysis.SpecialFunctions.Log.Basic
open Finset

Formal statement

namespace Vino

theorem norm_vmSum_le (α : ℝ) (N : ℕ) : ‖vmSum α N‖ ≤ psi 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 (the three primes theorem).

View graph

Trivial bound ∣S(α,N)∣≤ψ(N)|S(\alpha,N)|\le\psi(N)∣S(α,N)∣≤ψ(N)

Trivial bound $|S(\alpha,N)|\le\psi(N)$