Dirichlet Character

Overview

Dirichlet characters convert multiplicative residue-class information into complex-valued functions. They are the finite-group analogue of Fourier characters and provide a bridge from congruences to analytic number theory. Their values are constrained by multiplication, not assigned freely to each residue. The associated L-series organizes these periodic arithmetic values into an analytic function.

Prerequisites

Know the unit group modulo an integer, group homomorphisms, complex roots of unity and convergent series. A group character in this article is one-dimensional and complex-valued, not a trace character of an arbitrary higher-dimensional representation.

Definition and conventions

Fix $N\ge2$. A Dirichlet character modulo $N$ is a homomorphism

Extend it to all integers by periodicity on units and by $\chi(n)=0$ when $\gcd(n,N)>1$. Its nonzero values are roots of unity because the domain is finite. The extended function is completely multiplicative: $\chi(ab)=\chi(a)\chi(b)$ for all integers. The principal character has value one on every unit and zero on every nonunit.

The conductor is the least positive divisor $f$ of $N$ through whose unit group the character factors. A character is primitive when $f=N$. Factoring here refers to its values on units; extensions by zero for different moduli need not coincide on all integers.

Intuition

Characters are multiplicative probes of a finite group. Their complex phases distinguish residues while keeping multiplication simple. Zero extension excludes residues outside the unit group. Confusing the principal character with the constant-one function loses that exclusion and changes its L-function.

Examples and non-examples

Modulo $5$, the unit group is generated by $2$. Choosing $\chi(2)=i$ forces

Every multiple of $5$ has value zero. For example, $\chi(6)\chi(7)=1\cdot i=\chi(42)$. In contrast, declaring the value of every unit to be $-1$ fails at the identity, which any homomorphism must send to one. A general periodic function need not be a character.

Important properties

For a nonprincipal character, the sum over the unit group is zero. Choose a unit $u$ with $\chi(u)\ne1$. Multiplication by $u$ permutes the group, so the sum $S$ satisfies $S=\chi(u)S$, forcing $S=0$. Applying this to the quotient of two distinct characters gives their orthogonality.

For $\operatorname{Re}s>1$, the series $L(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s}$ converges absolutely. Complete multiplicativity and prime factorization give an Euler product. For the principal character modulo $N$, the product omits the primes dividing $N$, so it is not simply the Riemann zeta function without those factors being removed.

Connections and further reading

Character sums isolate arithmetic progressions, and primitive characters determine cleaner functional equations. Characters also appear in modular-form transformation laws. Zhang, Section 4.1.1 gives the definitions and analytic entry point.