Principal character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of ${\displaystyle \mathbb {Z} /k\mathbb {Z} }$. Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties. If ${\displaystyle \chi }$ is a Dirichlet character, one defines its Dirichlet L-series by

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.

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