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 ${\displaystyle s}$ a complex variable with real part greater than ${\displaystyle 1}$. 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; it is then called a Dirichlet L-function.

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in 1837 to prove his theorem on primes in arithmetic progressions. In his proof, Dirichlet showed that ${\displaystyle L(s,\chi )}$ is non-zero at ${\displaystyle s=1}$. Moreover, if ${\displaystyle \chi }$ is principal, then the corresponding Dirichlet L-function has a simple pole at ${\displaystyle s=1}$. Otherwise, the L-function is entire.