Dirichlet character: Difference between revisions

Latest revision as of 20:54, 6 May 2009

Definition

Let $n$ be a natural number. A Dirichlet character modulo $n$ is a function $χ : Z \to C$ such that:

$χ (a) = χ (a + n)$ for all $a \in Z$ ,
$χ (m) = 0$ whenever $m$ and $n$ are not relatively prime, and
for any $a, b \in N$ :

$χ (a b) = χ (a) χ (b)$ .

In other words, it is a homomorphism from the multiplicative monoid of the ring of integers to the ring of complex numbers that descends to a homomorphism from the ring of integers modulo $n$ .

If $n$ is the smallest period of $χ$ , and $χ$ is nonzero for all integers relatively prime to $n$ , then $χ$ is termed a primitive character modulo $n$ .

The all ones function is the trivial or principal character, and it is the only character with period $1$ .

Facts

Dirichlet characters are completely multiplicative: As part of the definition, any Dirichlet character is a completely multiplicative function. Thus, a Dirichlet character is completely determined by its value at prime numbers, and its Dirichlet series has an Euler product formula.

@@ Line 11: / Line 11: @@
 In other words, it is a homomorphism from the multiplicative monoid of the ring of integers to the ring of complex numbers that descends to a homomorphism from the ring of integers modulo <math>n</math>.
-If <math>n</math> is the smallest period of <math>\chi</math>, <math>\chi</math> is termed a primitive character modulo <math>n</math>.
+If <math>n</math> is the smallest period of <math>\chi</math>, and <math>\chi</math> is nonzero for all integers relatively prime to <math>n</math>, then <math>\chi</math> is termed a [[primitive character]] modulo <math>n</math>.
 The [[all ones function]] is the trivial or principal character, and it is the only character with period <math>1</math>.
+==Facts==
+* [[Dirichlet characters are completely multiplicative]]: As part of the definition, any Dirichlet character is a [[completely multiplicative function]]. Thus, a Dirichlet character is completely determined by its value at [[prime number]]s, and its [[Dirichlet series]] has an [[Euler product formula for Dirichlet series of completely multiplicative function|Euler product formula]].