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 $ℂ$ 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 $ℂ$ 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 $\mathrm{\Gamma }$, we get for $k$ positive, $k\equiv e_{\chi }$ mod 2,

$$L(\chi ,k)={(-1)}^{1+\frac{k-e_{\chi }}{2}}\frac{g_1(\chi )}{2i^{e_{\chi }}}{\left(\frac{2\pi }{m}\right)}^k\frac{B_{k,{\chi }^{-1}}}{k!}.$$(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.