PlanetMath: Dirichlet L-series

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Dirichlet L-series

(Definition)

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

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

(1)

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

$\displaystyle L(\chi,s)=\prod_p \frac{1}{1-\chi(p)p^{-s}}$

(2)

where the product is over all primes

, by virtue of the multiplicativity of $\chi$ . In the case where $\chi=\chi_0$ is the trivial character mod m, we have

$\displaystyle L(\chi_0,s)=\zeta(s)\prod_{p\vert m} (1-p^{-s}),$

(3)

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

$\displaystyle L(\chi,s)=L(\chi\prime,s)\prod_{p\vert m\atop p\nmid C_\chi}(1-\chi(p)p^{-s}),$

(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

$\displaystyle L(\chi,s)\left(\frac{m}{\pi}\right)^{s/2}\Gamma\left(\frac{s+e_\c... ...ft(\frac{m}{\pi}\right)^{\frac{1-s}{2}}\Gamma\left(\frac{1-s+e_\chi}{2}\right).$

(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

is a positive integer, we have

$\displaystyle L(\chi,1-k)=-\frac{B_{k,\chi}}{k},$

(6)

where $B_{k,\chi}$ is a generalized Bernoulli number. By (5), taking into account the poles of $\Gamma$ , we get for

positive, $k \equiv e_\chi$ mod 2,

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

(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.

"Dirichlet L-series" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

Dirichlet L-function

Attachments:: $L(\chi,1)\neq 0$ if $\chi$ is a real Dirichlet character (Theorem) by rm50

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Cross-references: number, complex analysis, Dirichlet's theorem on primes in arithmetic progression, poles, generalized Bernoulli number, integer, analytic, meromorphic continuation, Gauss sum, gamma function, functional equation, symmetric, analytic continuation, induces, primitive character, conductor, Riemann zeta function, trivial character, primes, product, identity, Euler product, positive, domain, converges absolutely, series, Dirichlet character
There are 3 references to this entry.

This is version 11 of Dirichlet L-series, born on 2003-01-20, modified 2006-10-25.
Object id is 3905, canonical name is DirichletLSeries.
Accessed 4765 times total.

Classification:

AMS MSC:

11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)

Pending Errata and Addenda

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