Dirichlet character

In number theory, a Dirichlet character is a multiplicative function on the positive integers which is derived from a character on the multiplicative group taken modulo a given integer.

Let N be a positive integer and write (Z/N)* for the multiplicative group of integers modulo N. Let χ be a group homomorphism from (Z/N)* to the unit circle. Since the multiplicative group is finite of order φ(N), where φ is the Euler totient function, the values of χ are all roots of unity. We extend χ to a function on the positive integers by defining χ(n) to be χ(n mod N) when n is coprime to N, and to be zero when n has a factor in common with N. This extended function is the Dirichlet character. As a function on the positive integers it is a totally multiplicative function with period n.

The principal character χ0 is derived from the trivial character which is 1 one n coprime to N and zero otherwise.

We say that a Dirichlet character χ1 with modulus N1 "induces" χ with modulus N if N1 divides N and χ(n) agrees with χ1(n) whenever they are both non-zero. A primitive character is one which is not induced from any character with smaller modulus. The conductor of a character is the modulus of the associated primitive character.

Dirichlet L-function
The Dirichlet L-function associated to χ is the Dirichlet series


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

with an Euler product


 * $$L(s,\chi) = \prod_p (1-\chi(p)p^{-s})^{-1} .\,$$

If χ is principal then L(s,χ) is the Riemann zeta function with finitely many Euler factors removed, and hence has a pole of order 1 at s=1. Otherwise L(s,χ) has a half-plane of convergence to the right of s=0. In all cases, L(s,χ) has an analytic continuation to the complex plane with a functional equation.