Empty prime exponential sum

Proved

Vino.primeSum_zero_len

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

analytic-number-theorycircle-methodnumber-theoryprime-numbers

For the unweighted prime exponential sum

T(\alpha,N)=\sum_{\substack{p<N\\ p\ \mathrm{prime}}}e(\alpha p),

the empty range gives $T(\alpha,0)=0$ .

Preamble

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

Formal statement

namespace Vino

theorem primeSum_zero_len (α : ℝ) : primeSum α 0 = 0 := 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).

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