Definition:General Dirichlet Series
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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:
- $\ds \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$
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
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.
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.
Sources
- 1915: G.H. Hardy and Marcel Riesz: The General Theory of Dirichlet's Series ... (next): $\text I$: Introduction $\S 1$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Dirichlet series