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Dirichlet character
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(Definition)
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A Dirichlet character modulo $m$ is a group homomorphism from $\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ to $\mathbb{C^*}$ . Dirichlet characters are usually denoted by the Greek letter $\chi$ . The function
is also referred to as a Dirichlet character. The Dirichlet characters modulo $m$ form a group if one defines $\chi\chi'$ to be the function which takes $a\in\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ to $\chi(a)\chi'(a)$ . It turns out that this resulting group is isomorphic to $\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ . The trivial character is given by $\chi(a)=1$ for all $a\in\left(\frac{\mathbb{Z}}{m\mathbb{Z}}\right)^*$ , and it acts as the identity element for the group. A character $\chi$ modulo $m$ is said to be induced by a character $\chi'$ modulo $m'$ if $m'\mid m$ and $\chi(n)=\chi'(n\bmod m')$ . A character which is not induced by any other character is called primitive.
If $\chi$ is non-primitive, the $\gcd$ of all such $m'$ is called the conductor of $\chi$ .
Examples:
- Legendre symbol $\legsym{n}{p}$ is a Dirichlet character modulo $p$ for any odd prime $p$ . More generally, Jacobi symbol $\legsym{n}{m}$ is a Dirichlet character modulo $m$ .
- The character modulo $4$ given by $\chi(1)=1$ and $\chi(3)=-1$ is a primitive character modulo $4$ . The only other character modulo $4$ is the trivial character.
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"Dirichlet character" is owned by bbukh. [ full author list (3) | owner history (2) ]
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Cross-references: Jacobi symbol, prime, odd, Legendre symbol, character, identity element, isomorphic, group, function, Greek letter, group homomorphism
There are 7 references to this entry.
This is version 7 of Dirichlet character, born on 2003-01-20, modified 2006-09-05.
Object id is 3906, canonical name is DirichletCharacter.
Accessed 12001 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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