Dirichlet Character - Axiomatic Definition

Axiomatic Definition

A Dirichlet character is any function χ from the integers to the complex numbers which has the following properties:

1. There exists a positive integer k such that χ(n) = χ(n + k) for all n.
2. If gcd(n,k) > 1 then χ(n) = 0; if gcd(n,k) = 1 then χ(n) ≠ 0.
3. χ(mn) = χ(m)χ(n) for all integers m and n.

From this definition, several other properties can be deduced. By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, k) = 1, property 2) says χ(1) ≠ 0, so

4. χ(1) = 1.

Properties 3) and 4) show that every Dirichlet character χ is completely multiplicative.

Property 1) says that a character is periodic with period k; we say that χ is a character to the modulus k. This is equivalent to saying that

5. If a ≡ b (mod k) then χ(a) = χ(b).

If gcd(a,k) = 1, Euler's theorem says that aφ(k) ≡ 1 (mod k) (where φ(k) is the totient function). Therefore by 5) and 4), χ(aφ(k)) = χ(1) = 1, and by 3), χ(aφ(k)) =χ(a)φ(k). So

6. For all a relatively prime to k, χ(a) is a φ(k)-th complex root of unity.

The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except the trivial one, which is 1 on all integers.

A character is called principal if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0. A character is called real if it assumes real values only. A character which is not real is called complex.

The sign of the character χ depends on its value at −1. Specifically, χ is said to be odd if χ(−1) = −1 and even if χ(−1) = 1.