Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form

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

where ${\displaystyle \chi }$ is a Dirichlet character and s a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire.