Dirichlet series

In mathematics, a Dirichlet series is any series of the form $${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},}$$ where s is complex, and ${\displaystyle a_{n}}$ is a complex sequence. It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.