Conjugating a complete character sum negates the phase

Proved

Weil.star_sum_addChar

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

exponential-sumsfinite-fieldsgauss-sumsnumber-theoryweil-bound

Let $F$ be a finite field, $\psi$ an additive character of $F$ valued in $\mathbb{C}$ and $g : F \to F$ any function. Then

\overline{\sum_{x \in F} \psi(g(x))} = \sum_{x \in F} \psi(-g(x)).

Preamble

import Mathlib.NumberTheory.GaussSum
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
import Mathlib.NumberTheory.MulChar.Lemmas
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.Index
import Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Algebra.Field.GeomSum
import Mathlib.Algebra.Order.BigOperators.Group.Finset

set_option autoImplicit false
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false

universe u_1 u_2

open AddChar MulChar Finset

Formal statement

namespace Weil
theorem star_sum_addChar : ∀ {F : Type u_1} [inst : Field F] [inst_1 : Fintype F] (ψ : AddChar F ℂ) (g : F → F), star (∑ x : F, (ψ : F → ℂ) (g x)) = ∑ x : F, (ψ : F → ℂ) (-g x) := by sorry
end Weil

Source

Iwaniec & Kowalski, Analytic Number Theory, AMS Colloquium Publications 53, 2004, Ch. 11 (Sums over finite fields), §11.1-11.3

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