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A residue-class indicator from character sums

Statement

Let $(a, q) = 1$ . Then for every integer $n$ ,

$\frac{1}{φ (q)} \sum_{χ m o d q} \overline{χ (a)} χ (n) = {\begin{cases} 1, & n \equiv a (m o d q), \\ 0, & n \equiv̸ a (m o d q) . \end{cases}$

Facts & Assumptions

Given: A reduced residue class $a$ modulo $q$ and an integer $n$ .

[L1]

For unit classes $u, v$ modulo $q$ , $\sum_{χ m o d q} χ (u) \overline{χ (v)} = φ (q)$ when $u = v$ and $0$ otherwise (Orthogonality relations for Dirichlet characters modulo q).

Proof

technique · direct

1.1L1givenalgebra

If $(n, q) > 1$ , then every Dirichlet character has $χ (n) = 0$ , so the sum is $0$ , and this matches the fact that $n$ cannot be congruent to the reduced class $a$ . If $(n, q) = 1$ , then both $a$ and $n$ are unit classes modulo $q$ and [L1] applies with $u = n$ and $v = a$ .

2.1step 1.1L1algebra∎

In the unit case from step 1.1, [L1] gives $\sum_{χ} \overline{χ (a)} χ (n) = φ (q)$ exactly when $n \equiv a (m o d q)$ , and $0$ otherwise. Dividing by $φ (q)$ yields the indicator formula.

Depends on

Orthogonality relations for Dirichlet characters modulo q

Used by

An orthogonality table for Dirichlet characters Example

Dependency tree · two levels

7 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

Kiran S. Kedlaya, Notes on Analytic Number Theory, Chapter 4 (standard reference, not scraped)
Leonard Tomczak, Analytic Number Theory, Corollary 4.3 (standard reference, not scraped)