TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Twisted Poisson summation

Statement

For primitive $χ$ modulo $q$ and Schwartz $f$ , $\sum_{n \in Z} χ (n) f (n) = \frac{τ (χ)}{q} \sum_{m \in Z} \overline{χ (m)} \hat{f} (m / q) .$

Facts & Assumptions

Given: A primitive character $χ$ modulo $q$ and a Schwartz function $f$ .

[F1]

Poisson summation holds for Schwartz functions (Poisson summation).

[F2]

The primitive additive twist is $\overline{χ (m)} τ (χ)$ (Primitive Gauss-sum twist).

Proof

technique · direct

1.1F1given

Decompose the left side by $n \equiv a (m o d q)$ and apply [F1] to each translated, $q$ -scaled Schwartz function.

2.1F2step 1.1algebra∎

The finite coefficient of $\hat{f} (m / q) / q$ is $\sum_{a m o d q} χ (a) e (a m / q)$ , which is [F2]. Substitution gives the formula.

Depends on

Used by

Even and odd theta kernels Example
Analytic continuation of primitive Dirichlet L-functions Theorem
Functional equation for primitive Dirichlet L-functions Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nickolas Andersen, Analytic Number Theory, Theorem 16.6 (standard reference, not scraped)