Dirichlet series

Remarks

Let $\sigma_a$ be the infimum of the real $\sigma$ such that $\sum_n \abs{a(n)} n^{-\sigma}$ converges, the abscissa of absolute convergence. The series converges absolutely for $\operatorname{Re} s > \sigma_a$ and defines a holomorphic function there. It also has an abscissa of convergence $\sigma_c$, with $\sigma_c \le \sigma_a \le \sigma_c + 1$. Example: for $a(n) = (-1)^{n+1}$, the series $\sum (-1)^{n+1} n^{-s}$ converges for $\operatorname{Re} s > 0$ and converges absolutely for $\operatorname{Re} s > 1$, so $\sigma_c = 0$ and $\sigma_a = 1$. Non-example: the power series $\sum_{n} a_n z^n$ is not a Dirichlet series, since its terms are powers $z^n$ of the variable rather than $n^{-s}$. If $F$ and $G$ are the Dirichlet series of $a$ and $b$, both absolutely convergent at $s$, then $F(s) G(s)$ is the Dirichlet series of the Dirichlet convolution $a * b$ at $s$. For example $\sum_n \mu(n) n^{-s} = 1/\zeta(s)$ for $\operatorname{Re} s > 1$, with $\mu$ the Möbius function.