General Dirichlet series

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of

${\displaystyle \sum _{n=1}^{\infty }a_{n}e^{-\lambda _{n}s},}$

where ${\displaystyle a_{n}}$, ${\displaystyle s}$ are complex numbers and ${\displaystyle \{\lambda _{n}\}}$ is a strictly increasing sequence of nonnegative real numbers that tends to infinity.

A simple observation shows that an 'ordinary' Dirichlet series

${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},}$

is obtained by substituting ${\displaystyle \lambda _{n}=\ln n}$ while a power series

${\displaystyle \sum _{n=1}^{\infty }a_{n}(e^{-s})^{n},}$

is obtained when ${\displaystyle \lambda _{n}=n}$.