Dirichlet L-series
The Dirichlet L-series associated to a Dirichlet character is the series
(1) |
It converges absolutely and uniformly in the domain for any positive , and admits the Euler product identity
(2) |
where the product is over all primes , by virtue of the multiplicativity of . In the case where is the trivial character mod m, we have
(3) |
where is the Riemann Zeta function. If is non-primitive, and is the conductor of , we have
(4) |
where is the primitive character which induces . For non-trivial, primitive characters mod m, admits an analytic continuation to all of and satsfies the symmetric functional equation
(5) |
Here, is defined by , is the gamma function, and is a Gauss sum. (3),(4), and (5) combined show that admits a meromorphic continuation to all of for all Dirichlet characters , and an analytic one for non-trivial . Again assuming that is non-trivial and primitive character mod m, if is a positive integer, we have
(6) |
where is a generalized Bernoulli number. By (5), taking into account the poles of , we get for positive, mod 2,
(7) |
This series was first investigated by Dirichlet (for whom they were named), who used the non-vanishing of for non-trivial 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 |