# Dirichlet Characters

## Motivation

The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as

$$
p\equiv a\pmod q.
$$

For example:

- Are there infinitely many primes congruent to $1\pmod4$?
- How are primes distributed modulo $q$?
- Do different residue classes contain approximately the same number of primes?

To study such questions analytically, entity["people","Peter Gustav Lejeune Dirichlet","German mathematician"] introduced Dirichlet characters.

These functions separate integers according to congruence behavior and form the basis of Dirichlet $L$-functions.

## Residue Classes Modulo $q$

Fix a positive integer $q$. Two integers $a$ and $b$ are congruent modulo $q$ if

$$
a\equiv b\pmod q.
$$

The invertible residue classes modulo $q$ form a finite multiplicative group:

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

Its elements are precisely the integers coprime to $q$, considered modulo $q$.

For example, modulo $5$,

$$
(\mathbb Z/5\mathbb Z)^\times =
\{1,2,3,4\}.
$$

Multiplication is performed modulo $5$.

## Definition of a Dirichlet Character

A Dirichlet character modulo $q$ is a function

$$
\chi:\mathbb Z\to\mathbb C
$$

satisfying:

### Periodicity

$$
\chi(n+q)=\chi(n)
$$

for all integers $n$.

### Multiplicativity

$$
\chi(mn)=\chi(m)\chi(n).
$$

### Vanishing on Nonunits

$$
\chi(n)=0
$$

whenever

$$
\gcd(n,q)>1.
$$

### Nonzero Values on Units

If

$$
\gcd(n,q)=1,
$$

then

$$
|\chi(n)|=1.
$$

Thus characters are periodic multiplicative functions taking values on the unit circle.

## Principal Character

The most important example is the principal character modulo $q$, denoted

$$
\chi_0.
$$

It is defined by

$$
\chi_0(n)=
\begin{cases}
1,& \gcd(n,q)=1,\\
0,& \gcd(n,q)>1.
\end{cases}
$$

This character behaves analogously to the constant function $1$ on invertible residue classes.

## Example Modulo $4$

Define

$$
\chi(n)=
\begin{cases}
0,& n\equiv0\pmod2,\\
1,& n\equiv1\pmod4,\\
-1,& n\equiv3\pmod4.
\end{cases}
$$

This is a Dirichlet character modulo $4$.

The values repeat periodically:

| $n$ | $\chi(n)$ |
|---|---|
| $1$ | $1$ |
| $2$ | $0$ |
| $3$ | $-1$ |
| $4$ | $0$ |
| $5$ | $1$ |

The function is multiplicative and distinguishes integers congruent to $1$ and $3$ modulo $4$.

## Group Structure

The set of Dirichlet characters modulo $q$ forms a finite abelian group under pointwise multiplication:

$$
(\chi_1\chi_2)(n)=\chi_1(n)\chi_2(n).
$$

The identity element is the principal character.

Every character has an inverse given by complex conjugation:

$$
\chi^{-1}(n)=\overline{\chi(n)}.
$$

The number of distinct characters modulo $q$ equals

$$
\varphi(q),
$$

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

## Characters as Homomorphisms

Dirichlet characters are precisely the group homomorphisms

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

Thus they play the role of Fourier characters for finite abelian groups.

This interpretation explains why characters are useful in harmonic analysis on arithmetic structures.

## Orthogonality Phenomena

Characters satisfy important cancellation identities.

For fixed $a$ coprime to $q$,

$$
\sum_{\chi \bmod q}\chi(a) =
\begin{cases}
\varphi(q),& a\equiv1\pmod q,\\
0,& \text{otherwise}.
\end{cases}
$$

Similarly,

$$
\sum_{a\bmod q}\chi(a) =
0
$$

for every nonprincipal character.

These orthogonality relations allow characters to isolate individual congruence classes.

## Detecting Congruence Classes

Characters can express arithmetic conditions analytically.

For example,

$$
\frac1{\varphi(q)}
\sum_{\chi\bmod q}
\overline{\chi(a)}\chi(n) =
\begin{cases}
1,& n\equiv a\pmod q,\\
0,& \text{otherwise}.
\end{cases}
$$

Thus characters behave like harmonic basis functions for modular arithmetic.

This identity is fundamental in the proof of Dirichlet’s theorem on primes in arithmetic progressions.

## Primitive Characters

Some characters arise from smaller moduli.

A character modulo $q$ is primitive if it does not factor through a proper divisor of $q$.

Primitive characters are the fundamental building blocks of Dirichlet $L$-functions and possess especially clean analytic properties.

## Importance in Number Theory

Dirichlet characters transform arithmetic congruence conditions into analytic multiplicative functions.

They provide the foundation for:

- Dirichlet $L$-functions,
- primes in arithmetic progressions,
- analytic class number formulas,
- modular forms,
- automorphic representations.

In modern number theory, characters serve as the simplest nontrivial examples of harmonic analysis on arithmetic groups.