Dirichlet character: Difference between revisions
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A Dirichlet character <math>\chi</math> is a periodic multiplicative function from the [[positive integer]]s to the [[real numbers]]. In mathematical notation we would say that a Dirichlet character is a function <math>\chi: \mathbb{Z} \to \mathbb{R}</math> such that | |||
1. <math>\chi(n + q) = \chi(n)</math> for all positive integers <math>n</math> and some integer q, and | |||
2. <math>\chi(mn) = \chi(m)\chi(n)</math> for all positive integers <math>m</math> and <math>n</math>. | |||
The smallest such <math>q</math> for which property 1 holds is known as the period of <math>\chi</math>. Typically we impose the additional restriction that <math>\chi(n) = 0</math> for all integers <math>n</math> such that <math>\gcd(n, q) = 0</math> where <math>q</math> is the period of <math>\chi</math>; with this restriction there are exactly <math>\phi(q)</math> such characters. | |||
The Dirichlet characters with period <math>q</math> have been completely classified. They are very useful in number theory. | |||
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[[Category: Number theory]] | |||
Latest revision as of 19:55, 2 September 2019
A Dirichlet character 
is a periodic multiplicative function from the positive integers to the real numbers. In mathematical notation we would say that a Dirichlet character is a function 
such that
1. 
for all positive integers 
and some integer q, and
2. 
for all positive integers 
and .
The smallest such 
for which property 1 holds is known as the period of . Typically we impose the additional restriction that 
for all integers such that 
where is the period of ; with this restriction there are exactly 
such characters.
The Dirichlet characters with period have been completely classified. They are very useful in number theory.
This article is a stub. Help us out by expanding it.
