# Orthogonality Relations

## Characters as Harmonic Functions

Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes modulo $q$.

The key mechanism behind this separation is orthogonality.

Orthogonality relations allow sums over characters to isolate specific congruence conditions. They are among the most important computational tools in analytic number theory.

## Characters Modulo $q$

Fix a positive integer $q$. Let

$$
\chi \pmod q
$$

denote a Dirichlet character modulo $q$.

The set of all characters modulo $q$ forms a finite abelian group under multiplication. The number of such characters equals

$$
\varphi(q),
$$

where $\varphi$ is Euler’s totient function.

The orthogonality relations describe cancellation among these characters.

## First Orthogonality Relation

Let $a$ be an integer coprime to $q$. Then

$$
\sum_{\chi \bmod q}\chi(a) =
\begin{cases}
\varphi(q),& a\equiv1\pmod q,\\
0,& a\not\equiv1\pmod q.
\end{cases}
$$

Thus the sum over all characters vanishes unless $a$ is the identity residue class modulo $q$.

This relation is analogous to the Fourier identity

$$
\sum_{k=0}^{n-1} e^{2\pi i km/n}=0
$$

unless

$$
m\equiv0\pmod n.
$$

## Proof Idea

Suppose first that

$$
a\equiv1\pmod q.
$$

Then every character satisfies

$$
\chi(a)=1,
$$

so

$$
\sum_{\chi\bmod q}\chi(a)=\varphi(q).
$$

Now suppose

$$
a\not\equiv1\pmod q.
$$

Because the character group is finite, multiplication by the fixed value $\chi(a)$ permutes the character values nontrivially. This forces complete cancellation in the sum.

Hence the total is zero.

## Second Orthogonality Relation

Fix a character $\chi$. Then

$$
\sum_{a\bmod q}\chi(a) =
\begin{cases}
\varphi(q),& \chi=\chi_0,\\
0,& \chi\neq\chi_0,
\end{cases}
$$

where $\chi_0$ is the principal character.

Thus every nonprincipal character averages to zero over the reduced residue classes modulo $q$.

This cancellation property is fundamental in analytic estimates.

## Proof of the Second Relation

If

$$
\chi=\chi_0,
$$

then

$$
\chi(a)=1
$$

for every residue class coprime to $q$, so the sum equals

$$
\varphi(q).
$$

Suppose now that $\chi\neq\chi_0$. Then there exists some $b$ with

$$
\chi(b)\neq1.
$$

Multiplication by $b$ permutes the reduced residue classes modulo $q$, giving

$$
\sum_{a\bmod q}\chi(a) =
\sum_{a\bmod q}\chi(ab) =
\chi(b)\sum_{a\bmod q}\chi(a).
$$

Since

$$
\chi(b)\neq1,
$$

the sum must vanish.

## Detecting Congruence Classes

The orthogonality relations allow exact detection of congruence conditions.

For integers $a,n$ coprime to $q$,

$$
\frac1{\varphi(q)}
\sum_{\chi\bmod q}
\overline{\chi(a)}\chi(n) =
\begin{cases}
1,& n\equiv a\pmod q,\\
0,& n\not\equiv a\pmod q.
\end{cases}
$$

This identity is one of the most important formulas in analytic number theory.

It converts congruence restrictions into sums over characters.

## Example Modulo $4$

Modulo $4$, there are two characters:

- the principal character $\chi_0$,
- the nontrivial character $\chi$.

Their values on odd residues are:

| $n$ | $\chi_0(n)$ | $\chi(n)$ |
|---|---|---|
| $1$ | $1$ | $1$ |
| $3$ | $1$ | $-1$ |

Then

$$
\frac12(\chi_0(n)+\chi(n)) =
\begin{cases}
1,& n\equiv1\pmod4,\\
0,& n\equiv3\pmod4.
\end{cases}
$$

Thus characters isolate individual residue classes through cancellation.

## Fourier-Analytic Interpretation

Dirichlet characters form an orthogonal basis for functions on the finite group

$$
(\mathbb Z/q\mathbb Z)^\times.
$$

Every function on this group can therefore be expanded as a linear combination of characters.

This is the finite-group analogue of Fourier series expansions.

The coefficients in such expansions are computed using orthogonality relations.

## Applications to Prime Numbers

Orthogonality relations are central in Dirichlet’s theorem on arithmetic progressions.

To study primes satisfying

$$
p\equiv a\pmod q,
$$

one writes the congruence condition using characters:

$$
1_{p\equiv a\pmod q} =
\frac1{\varphi(q)}
\sum_{\chi\bmod q}
\overline{\chi(a)}\chi(p).
$$

The problem then becomes the study of sums involving characters and Dirichlet $L$-functions.

Thus orthogonality transforms arithmetic constraints into analytic expressions.

## Importance

Orthogonality relations are the basic computational engine of character theory.

They provide:

- detection of congruence classes,
- Fourier decomposition on finite groups,
- cancellation identities,
- analytic access to arithmetic progressions.

These relations underlie much of analytic number theory, harmonic analysis on groups, and modern automorphic theory.