Definition:General Dirichlet Series

Definition
Let $a_n$ be a sequence in $\C$.

Let $\sequence {\lambda_n}$ be a strictly increasing sequence of non-negative real numbers whose limit is infinity.

A general Dirichlet series of type $\lambda_n$ is a complex function $f: \C \to \C$ defined by the series:


 * $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$

which is defined at the points where it converges.

Notation
It is a historical convention that the variable $s$ is written $s = \sigma + i t$ with $\sigma, t \in \R$.

Also known as
Some treatments of this subject use the possessive style: general Dirichlet's series.

However, that makes it sound as though it is named after a high-ranking military officer.

Also see

 * Definition:Abscissa of Convergence
 * Definition:Abscissa of Absolute Convergence

Examples

 * An ordinary Dirichlet series is the case when $\lambda_n = \map \log n$.
 * Setting $\lambda_n = n$ gives a power series of exponential type.