Dirichlet L-function

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

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

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.

Zeros of the Dirichlet L-functions

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

$\beta < 1 - \frac{c}{ \log \big(q(2+|\gamma|)\big)} \$

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 product

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:

$L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,$

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

Functional equation

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

$\Lambda(s,\chi) = \left(\frac{\pi}{k}\right)^{-(s+a)/2} \Gamma\left(\frac{s+a}{2}\right) L(s,\chi),$

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

$a=\begin{cases}0;&\mbox{if }\chi(-1)=1, \\ 1;&\mbox{if }\chi(-1)=-1,\end{cases}$

one has the functional equation

$\Lambda(1-s,\overline{\chi})=\frac{i^ak^{1/2}}{\tau(\chi)}\Lambda(s,\chi).$

Here we wrote τ(χ) for the Gauss sum

$\sum_{n=1}^k\chi(n)\exp(2\pi in/k).$

Note that |τ(χ)| = k1/2.

Relation to the Hurwitz zeta-function

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

$L(s,\chi) = \sum_{n=1}^\infty \frac {\chi(n)}{n^s} = \frac {1}{k^s} \sum_{m=1}^k \chi(m)\; \zeta \left(s,\frac{m}{k}\right).$

In particular, the Dirichlet L-function of the trivial character (which implies the modulus k is prime) yields the Riemann zeta-function:

$\zeta(s) = \frac {1}{k^s} \sum_{m=1}^k \zeta \left(s,\frac{m}{k}\right).$