Dirichlet series in the context of "Analytic number theory"

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. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the constant unit function u(n), namely: