Dirichlet character

A periodic, completely multiplicative arithmetic function that vanishes on integers not coprime to its modulus and restricts to a group character on the units modulo that modulus.

Group character: A homomorphism from a group to the multiplicative group of nonzero complex numbers. On the units modulo its modulus, a Dirichlet character is precisely such a homomorphism.
Dirichlet character modulo n: A Dirichlet character whose periodicity and unit group are specified by the positive integer n. Fixing n determines which integers vanish and which residue classes carry character values.
Dirichlet L-function: The series L(s,χ)=Σχ(n)n⁻ˢ, initially defined for complex s with real part greater than 1. A Dirichlet character supplies the coefficients that encode arithmetic in an L-function.
Additive character: A homomorphism from an additive group to the multiplicative group of complex numbers of absolute value 1. It encodes addition rather than multiplication on residue classes.
Unit group modulo n: The group of residue classes modulo n that are coprime to n, under multiplication. This is the finite group on which a Dirichlet character takes its nonzero values.
Conductor (number theory): The least modulus from which a Dirichlet character is induced. It identifies the smallest modulus that captures the character's essential periodic behavior.
Dirichlet's theorem on arithmetic progressions: The theorem that every reduced residue class modulo n contains infinitely many primes. Dirichlet characters are the central analytic tool in its proof.
Legendre symbol: A function modulo an odd prime that records whether a residue is zero, a square, or a nonsquare. It is a familiar quadratic Dirichlet character modulo an odd prime.
Euler's totient function: The function φ(n) counts the integers from 1 to n that are coprime to n. It gives the order of the unit group modulo n and thus the number of character values to organize.
Primitive character: A Dirichlet character that is not induced from a character of any proper divisor of its modulus. Primitive characters retain the full information of their stated modulus.

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