# Dirichlet L-series

The Dirichlet L-series associated to a Dirichlet character $\chi$ is the series

 $L(\chi,s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}.$ (1)

It converges absolutely and uniformly in the domain $\Re(s)\geq 1+\delta$ 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

 $L(\chi,s)=L(\chi\prime,s)\prod_{p|m\atop p\nmid C_{\chi}}(1-\chi(p)p^{-s}),$ (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 $\mathbb{C}$ and satsfies the symmetric functional equation

 $L(\chi,s)\left(\frac{m}{\pi}\right)^{s/2}\Gamma\left(\frac{s+e_{\chi}}{2}% \right)=\frac{g_{1}(\chi)}{i^{e_{\chi}}\sqrt{m}}L(\chi^{-1},1-s)\left(\frac{m}% {\pi}\right)^{\frac{1-s}{2}}\Gamma\left(\frac{1-s+e_{\chi}}{2}\right).$ (5)

Here, $e_{\chi}\in\{0,1\}$ is defined by $\chi(-1)=(-1)^{e_{\chi}}\chi(1)$, $\Gamma$ 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 $\mathbb{C}$ 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 $\Gamma$, we get for $k$ positive, $k\equiv e_{\chi}$ mod 2,

 $L(\chi,k)=(-1)^{1+\frac{k-e_{\chi}}{2}}\frac{g_{1}(\chi)}{2i^{e_{\chi}}}\left(% \frac{2\pi}{m}\right)^{k}\frac{B_{k,\chi^{-1}}}{k!}.$ (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 DirichletLseries 2013-03-22 13:22:28 2013-03-22 13:22:28 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 11M06 Dirichlet L-function LSeriesOfAnEllipticCurve DirichletSeries