Difference between revisions of "Dirichlet character"

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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
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1. <math>\chi(n + q) = \chi(n)</math> for all positive integers <math>n</math> and some integer q, and
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2. <math>\chi(mn) = \chi(m)\chi(n)</math> for all positive integers <math>m</math> and <math>n</math>.
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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.
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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 20:55, 2 September 2019

A Dirichlet character $\chi$ 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 $\chi: \mathbb{Z} \to \mathbb{R}$ such that

1. $\chi(n + q) = \chi(n)$ for all positive integers $n$ and some integer q, and 2. $\chi(mn) = \chi(m)\chi(n)$ for all positive integers $m$ and $n$.

The smallest such $q$ for which property 1 holds is known as the period of $\chi$. Typically we impose the additional restriction that $\chi(n) = 0$ for all integers $n$ such that $\gcd(n, q) = 0$ where $q$ is the period of $\chi$; with this restriction there are exactly $\phi(q)$ such characters.

The Dirichlet characters with period $q$ have been completely classified. They are very useful in number theory.

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