Dirichlet L-function: Difference between revisions

Latest revision as of 13:51, 7 May 2009

Definition

A Dirichlet L-function is a meromorphic function obtained as the analytic continuation of the Dirichlet series of the Dirichlet character.

The corresponding Dirichlet series is termed a Dirichlet L-series, and the Dirichlet L-series as well as the function are denoted $L(s,\chi )$ where $\chi$ is the character and $s$ is the argument.

$L(s,\chi ):=\sum _{n\in \mathbb {N} }{\frac {\chi (n)}{n^{s}}}=\prod _{p\in \mathbb {P} }{\frac {1}{1-\chi (p)p^{-s}}}$ ,

where $\mathbb {P}$ is the set of all prime numbers. The equality follows from the Euler product formula.

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 A '''Dirichlet L-function''' is a [[meromorphic function]] obtained as the analytic continuation of the [[defining ingredient::Dirichlet series]] of the [[defining ingredient::Dirichlet character]].
-The corresponding Dirichlet series is termed a Dirichlet L-series, and the Dirichlet L-series as well as the function are denoted <math>L(\chi,s)</math> where <math>\chi</math> is the character and <math>s</math> is the argument.
+The corresponding Dirichlet series is termed a Dirichlet L-series, and the Dirichlet L-series as well as the function are denoted <math>L(s,\chi)</math> where <math>\chi</math> is the character and <math>s</math> is the argument.
-<math>L(\chi,s) := \sum_{n \in \mathbb{N}} \frac{\chi(n)}{n^s} = \prod_{p \in \mathbb{P}} \frac{1}{1 - \chi(p)p^{-s}}</math>,
+<math>L(s,\chi) := \sum_{n \in \mathbb{N}} \frac{\chi(n)}{n^s} = \prod_{p \in \mathbb{P}} \frac{1}{1 - \chi(p)p^{-s}}</math>,
 where <math>\mathbb{P}</math> is the set of all [[prime number]]s. The equality follows from the [[Euler product formula for Dirichlet series of completely multiplicative function|Euler product formula]].