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. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the constant unit function u(n), namely: ${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\sum _{n=1}^{\infty }{\frac {u(n)}{n^{s}}}=D(u,s),}$ where D(u, s) denotes the Dirichlet series of u(n). 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.