Open main menu

In mathematics, a Dirichlet L-series is a function of the form

Here χ is a Dirichlet character and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1.

Contents

Zeros of the Dirichlet L-functionsEdit

If χ is a primitive character with χ(−1) = 1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative even integers. If χ is a primitive character with χ(−1) = −1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative odd integers.

Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions: for example, for χ a non-real character of modulus q, we have

 

for β + iγ a non-real zero.[1]

Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the Dirichlet L-functions are conjectured to obey the generalized Riemann hypothesis.

Euler productEdit

Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:

 

where the product is over all prime numbers.[2]

Functional equationEdit

Let us assume that χ is a primitive character to the modulus k. Defining

 

where Γ denotes the Gamma function and the symbol a is given by

 

one has the functional equation

 

Here we wrote τ(χ) for the Gauss sum

 

Note that |τ(χ)| = k1/2.

Relation to the Hurwitz zeta-functionEdit

The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta-function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as

 

See alsoEdit

NotesEdit

  1. ^ Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society. p. 163. ISBN 0-8218-0737-4. Zbl 0814.11001.
  2. ^ Apostol 1976, Theorem 11.7

ReferencesEdit