# Dirichlet $L$-Functions

## Generalizing the Zeta Function

The Riemann zeta function

$$
\zeta(s) =
\sum_{n=1}^{\infty}\frac1{n^s}
$$

studies primes globally, without distinguishing residue classes modulo $q$.

To study primes in arithmetic progressions, entity["people","Peter Gustav Lejeune Dirichlet","German mathematician"] introduced a family of functions built from Dirichlet characters.

These are the Dirichlet $L$-functions.

They are among the most important objects in analytic number theory and provide the analytic foundation for Dirichlet’s theorem on primes in arithmetic progressions.

## Definition

Let

$$
\chi \pmod q
$$

be a Dirichlet character modulo $q$.

For

$$
\operatorname{Re}(s)>1,
$$

the Dirichlet $L$-function associated to $\chi$ is

$$
L(s,\chi) =
\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}.
$$

This is a Dirichlet series weighted by the character values.

When $\chi=\chi_0$ is the principal character, the series resembles the zeta function.

## Examples

### Principal Character Modulo $q$

If $\chi_0$ is the principal character modulo $q$, then

$$
L(s,\chi_0) =
\sum_{\substack{n\geq1\\(n,q)=1}}
\frac1{n^s}.
$$

This equals

$$
L(s,\chi_0) =
\zeta(s)
\prod_{p\mid q}
\left(1-\frac1{p^s}\right).
$$

Thus the principal $L$-function differs from the zeta function only by finitely many Euler factors.

### Nontrivial Character Modulo $4$

Consider the character

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

Then

$$
L(s,\chi) =
1-\frac1{3^s}+\frac1{5^s}-\frac1{7^s}+\cdots.
$$

At $s=1$, this becomes the alternating Leibniz series:

$$
L(1,\chi) =
1-\frac13+\frac15-\frac17+\cdots =
\frac{\pi}{4}.
$$

## Euler Product

Because characters are multiplicative, the Dirichlet series factors into an Euler product:

$$
L(s,\chi) =
\prod_p
\left(1-\frac{\chi(p)}{p^s}\right)^{-1},
\qquad
\operatorname{Re}(s)>1.
$$

This formula follows exactly as for the zeta function.

The Euler product shows that $L$-functions encode prime-number information filtered by congruence conditions.

## Convergence

For

$$
\operatorname{Re}(s)>1,
$$

the series converges absolutely because

$$
|\chi(n)|\leq1.
$$

Indeed,

$$
\sum_{n=1}^{\infty}
\left|
\frac{\chi(n)}{n^s}
\right|
\leq
\sum_{n=1}^{\infty}\frac1{n^\sigma},
$$

where

$$
\sigma=\operatorname{Re}(s)>1.
$$

Absolute convergence justifies termwise manipulations and Euler products.

## Analytic Continuation

Like the zeta function, Dirichlet $L$-functions extend analytically beyond their initial region of convergence.

If $\chi\neq\chi_0$ is nonprincipal, then

$$
L(s,\chi)
$$

extends to an entire function on the complex plane.

If $\chi=\chi_0$, then $L(s,\chi_0)$ has a simple pole at

$$
s=1,
$$

inherited from the zeta function.

This distinction between principal and nonprincipal characters is crucial in Dirichlet’s theorem.

## Functional Equations

Primitive characters give rise to completed $L$-functions satisfying functional equations analogous to that of the zeta function.

After suitable normalization, one obtains identities relating values at $s$ and $1-s$.

These equations reveal deep symmetries and allow analytic study inside the critical strip.

## Nonvanishing at $s=1$

The key theorem behind primes in arithmetic progressions is:

If $\chi\neq\chi_0$, then

$$
L(1,\chi)\neq0.
$$

This nonvanishing result is the heart of Dirichlet’s theorem.

It prevents excessive cancellation among primes in a fixed residue class.

## Dirichlet’s Theorem

Using characters and orthogonality relations, one can isolate primes satisfying

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

Dirichlet proved:

If

$$
(a,q)=1,
$$

then there are infinitely many primes congruent to $a$ modulo $q$.

The proof relies on logarithmic divergence associated with the principal character and boundedness of the nonprincipal $L$-functions near $s=1$.

Thus $L$-functions provide analytic access to arithmetic progressions of primes.

## General Philosophy

Dirichlet $L$-functions illustrate a central principle of modern number theory:

arithmetic symmetry produces analytic structure.

The character $\chi$ encodes congruence information, while the analytic properties of $L(s,\chi)$ reveal distribution laws for primes.

This philosophy extends broadly:

| Arithmetic Data | Analytic Object |
|---|---|
| residue classes | Dirichlet characters |
| primes in progressions | Dirichlet $L$-functions |
| number fields | Dedekind zeta functions |
| modular forms | automorphic $L$-functions |

Dirichlet $L$-functions are therefore the first major example of the modern $L$-function framework.

## Importance

Dirichlet $L$-functions generalize the zeta function while preserving its essential analytic structure:

- Dirichlet series,
- Euler products,
- analytic continuation,
- functional equations,
- zero distributions.

They connect harmonic analysis on finite groups with prime-number distribution and form one of the cornerstones of analytic number theory.