Dirichlet L-series

The Dirichlet L-series associated to a Dirichlet characterDlmfMathworldPlanetmath χ is the series

L(χ,s)=n=1χ(n)ns. (1)

It converges absolutely and uniformly in the domain (s)1+δ for any positive δ, and admits the Euler productMathworldPlanetmath identity

L(χ,s)=p11-χ(p)p-s (2)

where the product is over all primes p, by virtue of the multiplicativity of χ. In the case where χ=χ0 is the trivial character mod m, we have

L(χ0,s)=ζ(s)p|m(1-p-s), (3)

where ζ(s) is the Riemann Zeta functionDlmfDlmfMathworldPlanetmath. If χ is non-primitive, and Cχ is the conductor of χ, we have

L(χ,s)=L(χ,s)p|mpCχ(1-χ(p)p-s), (4)

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

L(χ,s)(mπ)s/2Γ(s+eχ2)=g1(χ)ieχmL(χ-1,1-s)(mπ)1-s2Γ(1-s+eχ2). (5)

Here, eχ{0,1} is defined by χ(-1)=(-1)eχχ(1), Γ is the gamma functionDlmfDlmfMathworldPlanetmath, and g1(χ) is a Gauss sumDlmfPlanetmath. (3),(4), and (5) combined show that L(χ,s) admits a meromorphic continuation to all of for all Dirichlet characters χ, and an analytic one for non-trivial χ. Again assuming that χ is non-trivial and primitive character mod m, if k is a positive integer, we have

L(χ,1-k)=-Bk,χk, (6)

where Bk,χ is a generalized Bernoulli numberDlmfPlanetmath. By (5), taking into account the poles of Γ, we get for k positive, keχ mod 2,

L(χ,k)=(-1)1+k-eχ2g1(χ)2ieχ(2πm)kBk,χ-1k!. (7)

This series was first investigated by Dirichlet (for whom they were named), who used the non-vanishing of L(χ,1) for non-trivial χ 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