# Dirichlet L-series

The *Dirichlet L-series* associated to a Dirichlet character^{} $\chi $ is the series

$$L(\chi ,s)=\sum _{n=1}^{\mathrm{\infty}}\frac{\chi (n)}{{n}^{s}}.$$ | (1) |

It converges absolutely and uniformly in the domain $\mathrm{\Re}(s)\ge 1+\delta $ for any positive $\delta $, and admits the Euler product^{} identity

$$L(\chi ,s)=\prod _{p}\frac{1}{1-\chi (p){p}^{-s}}$$ | (2) |

where the product is over all primes $p$, by virtue of the multiplicativity of $\chi $. In the case where $\chi ={\chi}_{0}$ is the trivial character mod m, we have

$$L({\chi}_{0},s)=\zeta (s)\prod _{p|m}(1-{p}^{-s}),$$ | (3) |

where $\zeta (s)$ is the Riemann Zeta function^{}. If $\chi $ is non-primitive, and ${C}_{\chi}$ is the conductor of $\chi $, we have

$$L(\chi ,s)=L(\chi \prime ,s)\prod _{\genfrac{}{}{0pt}{}{p|m}{p\nmid {C}_{\chi}}}(1-\chi (p){p}^{-s}),$$ | (4) |

where $\chi \prime $ is the primitive character which induces $\chi $. For non-trivial, primitive characters $\chi $ mod m, $L(\chi ,s)$ admits an analytic continuation to all of $\u2102$ and satsfies the symmetric functional equation

$$L(\chi ,s){\left(\frac{m}{\pi}\right)}^{s/2}\mathrm{\Gamma}\left(\frac{s+{e}_{\chi}}{2}\right)=\frac{{g}_{1}(\chi )}{{i}^{{e}_{\chi}}\sqrt{m}}L({\chi}^{-1},1-s){\left(\frac{m}{\pi}\right)}^{\frac{1-s}{2}}\mathrm{\Gamma}\left(\frac{1-s+{e}_{\chi}}{2}\right).$$ | (5) |

Here, ${e}_{\chi}\in \{0,1\}$ is defined by $\chi (-1)={(-1)}^{{e}_{\chi}}\chi (1)$, $\mathrm{\Gamma}$ is the gamma function^{}, and ${g}_{1}(\chi )$ is a Gauss sum^{}.
(3),(4), and (5) combined show that $L(\chi ,s)$ admits a meromorphic continuation to all of $\u2102$ for all Dirichlet characters $\chi $, and an analytic one for non-trivial $\chi $.
Again assuming that $\chi $ is non-trivial and primitive character mod m, if $k$ is a positive integer, we have

$$L(\chi ,1-k)=-\frac{{B}_{k,\chi}}{k},$$ | (6) |

where ${B}_{k,\chi}$ is a generalized Bernoulli number^{}. By (5), taking into account the poles of $\mathrm{\Gamma}$, we get for $k$ positive, $k\equiv {e}_{\chi}$ mod 2,

$$L(\chi ,k)={(-1)}^{1+\frac{k-{e}_{\chi}}{2}}\frac{{g}_{1}(\chi )}{2{i}^{{e}_{\chi}}}{\left(\frac{2\pi}{m}\right)}^{k}\frac{{B}_{k,{\chi}^{-1}}}{k!}.$$ | (7) |

This series was first investigated by Dirichlet (for whom they were named), who used the non-vanishing of $L(\chi ,1)$ for non-trivial $\chi $ to prove his famous Dirichlet’s theorem on primes in arithmetic progression. This is probably the first instance of using complex analysis to prove a purely number theoretic result.

Title | Dirichlet L-series |
---|---|

Canonical name | DirichletLseries |

Date of creation | 2013-03-22 13:22:28 |

Last modified on | 2013-03-22 13:22:28 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 14 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 11M06 |

Synonym | Dirichlet L-function |

Related topic | LSeriesOfAnEllipticCurve |

Related topic | DirichletSeries |