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.