Dirichlet L-function

In mathematics, a Dirichlet L-series, named in honour of Johann Peter Gustav Lejeune Dirichlet, 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). An important special case of a Dirichlet L-function, namely the one in which χ is the trivial character, is the Riemann zeta function.

It was proven by Dirichlet that L(1,χ)≠0 for all Dirichlet characters χ, allowing him to establish his theorem on primes in arithmetic progressions. 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.

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.

Functional equation

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

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

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

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

Here we wrote τ(χ) for the Gauss sum

$\sum _{n=1}^{k}\chi (n)\exp(2\pi im/q).$

Note that |τ(χ)|=k^1/2.

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 $\chi$ be a character modulo k. Then we can write its Dirichlet L-function as

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

In particular, the Dirichlet L-function of the trivial character modulo 1 yields the Riemann zeta-function:

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