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.