# Dirichlet character

Dirichlet introduced these functions in his 1837 paper on primes in arithmetic progressions.^{[3]}^{[4]}

There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of (introduced by Brian Conrey and used by the ).

Paraphrasing Davenport^{[10]} Dirichlet characters can be regarded as a particular case of Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.

10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group.

The first factor is not zero, therefore the second one is. Since the relations are equivalent, the second one is also proved. QED

The second relation can be proven directly in the same way, but requires a lemma^{[24]}

Any character mod a prime power is also a character mod every larger power. For example, mod 16^{[27]}

A related phenomenon can happen with a character mod the product of primes; its *nonzero values* may be periodic with a smaller period.

Primitive characters often simplify (or make possible) formulas in the theories of L-functions^{[30]} and modular forms.

This distinction appears in the functional equation of the Dirichlet L-function.

If the modulus is the absolute value of a fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters^{[29]} they are imaginary.^{[37]}

Dirichlet characters appear several places in the theory of modular forms and functions. A typical example is^{[38]}

It is not necessary to establish the defining properties 1) - 3) to show that a function is a Dirichlet character.

Lists 30,397,486 Dirichlet characters of modulus up to 10,000 and their L-functions