Dirichlet series

Summary

A Dirichlet series is a series $\sum_{n \ge 1} a(n) n^{-s}$, where $a$ is an arithmetic function and $s$ is a complex variable. It converges in a right half-plane and defines a holomorphic function there. Multiplying two Dirichlet series corresponds to the Dirichlet convolution of their coefficients, which makes them the natural generating functions of multiplicative number theory. The Riemann zeta function and the Dirichlet L-functions are the main examples.