Dirichlet L-function

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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.

[edit] 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.

[edit] 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

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

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.

[edit] 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