Dirichlet Characters and Group Characters mod n

The precise definition

Fix a modulus n ≥ 1. A Dirichlet character modulo n is a function χ: ℤ → ℂ satisfying three conditions:

• Periodicity: χ(a) = χ(b) whenever a ≡ b (mod n).
• Complete multiplicativity: χ(ab) = χ(a)χ(b) for all a, b ∈ ℤ.
• Support on units: χ(a) ≠ 0 if and only if gcd(a, n) = 1.

Equivalently — and this is the cleaner statement — χ is a group homomorphism from the unit group (ℤ/nℤ)ˣ to the multiplicative group ℂˣ, extended to all of ℤ by declaring χ(a) = 0 when gcd(a, n) > 1. Because (ℤ/nℤ)ˣ is a finite group of order φ(n), every value χ(a) with gcd(a,n)=1 is a φ(n)-th root of unity: χ(a)^{φ(n)} = χ(a^{φ(n)}) = χ(1) = 1, by Euler's theorem. So a Dirichlet character takes values on the unit circle and at 0. The principal character χ₀ sends every unit to 1 (and non-units to 0); it is the identity of the character group.

The picture: characters as pure frequencies

Think of functions on the finite group G = (ℤ/nℤ)ˣ as signals, and characters as the pure tones. Just as e^{2πikx} form an orthogonal basis for periodic functions on ℝ, the characters χ form an orthonormal basis (up to the factor 1/|G|) for all class functions on G. Since G is abelian, every irreducible representation is one-dimensional, and these one-dimensional representations are the characters.

The set of all Dirichlet characters mod n, under pointwise multiplication (χ·ψ)(a) = χ(a)ψ(a), forms a group Ĝ called the dual group or character group. A cornerstone fact: for a finite abelian group, Ĝ ≅ G — non-canonically, but genuinely isomorphic. Hence there are exactly φ(n) characters mod n. The picture to keep: characters spread the arithmetic of residue classes across the unit circle, so that a linear combination of them can spotlight any single class a while cancelling all others.

The mechanism: orthogonality relations

The engine that makes characters useful is a pair of orthogonality relations. Summing a character over a complete set of units gives:

• ∑_{a mod n} χ(a) = φ(n) if χ = χ₀, and 0 otherwise (sum over a).
• ∑_{χ} χ(a) = φ(n) if a ≡ 1 (mod n), and 0 otherwise (sum over all φ(n) characters).

Why the first holds: if χ ≠ χ₀, pick b with χ(b) ≠ 1. As a runs over the units, so does ab, so S = ∑χ(a) = ∑χ(ab) = χ(b)·S. Thus (χ(b) − 1)S = 0, forcing S = 0. This is the classic shift trick. The second relation follows from the isomorphism Ĝ ≅ G by symmetry. Combining them yields the indicator formula: for gcd(a, n) = 1, the class of a is detected by (1/φ(n)) ∑_{χ} χ̄(a) χ(m), which equals 1 when m ≡ a (mod n) and 0 otherwise. This is the discrete Fourier inversion that sieves a single progression out of the integers.

Worked example: characters mod 5 and mod 8

Modulo 5. The group (ℤ/5ℤ)ˣ = {1,2,3,4} is cyclic of order 4, generated by g = 2 (powers: 2, 4, 3, 1). A character is determined by its value χ(2), which must be a 4th root of unity: 1, i, −1, or −i. These four choices give the four characters, matching φ(5) = 4. The choice χ(2) = i produces an order-4 character; χ(2) = −1 gives the quadratic (Legendre) character, which equals the Legendre symbol (a|5); χ(2) = 1 is principal.

Modulo 8. Here (ℤ/8ℤ)ˣ = {1,3,5,7} ≅ ℤ/2 × ℤ/2 is not cyclic, so there is no single generator. Its four characters all take values in {±1}. Writing them out, one recovers the sign patterns behind the Jacobi symbols and the quadratic-reciprocity supplements: the character controlling when 2 is a quadratic residue is χ(a)=(−1)^{(a²−1)/8}, with χ(1)=1, χ(3)=−1, χ(5)=−1, χ(7)=1. The character with χ(3)=−1, χ(5)=1 is instead (−1)^{(a−1)/2}, induced from mod 4 and unrelated to the residue status of 2. This shows characters need not be powers of one generator — the structure follows the group's decomposition.

Why the hypotheses matter; primitivity and conductor

Drop complete multiplicativity and you no longer have a homomorphism, so orthogonality collapses and the indicator formula fails — the whole detection machinery rests on χ(ab)=χ(a)χ(b). Drop the support-on-units convention and periodicity becomes inconsistent with multiplicativity (a non-unit residue can equal a product of units), so setting χ = 0 there is forced, not optional.

A subtler issue is induced characters. If d | n and ψ is a character mod d, it induces a character mod n by χ(a) = ψ(a) for gcd(a,n)=1 (else 0). A character is primitive if it is not induced from any proper divisor; the smallest such d is its conductor. This distinction is essential: the functional equation of a Dirichlet L-function, and the correct Gauss-sum evaluation |τ(χ)| = √d, hold only for primitive χ. Non-primitive characters carry spurious Euler factors that must be stripped. Connections run to Gauss sums, quadratic reciprocity, class field theory (via the Kronecker–Weber theorem), and Tate's thesis, where characters become the local–global Hecke characters.

Significance: L-functions and primes in progressions

The payoff is Dirichlet's L-function: L(s, χ) = ∑_{m≥1} χ(m)/mˢ = ∏_p (1 − χ(p)p^{−s})^{−1} for Re(s) > 1, an Euler product because χ is multiplicative. Taking the logarithmic derivative and applying the orthogonality indicator to weight primes p ≡ a (mod n) yields ∑_{p ≡ a} p^{−s} ≈ (1/φ(n)) ∑_χ χ̄(a) log L(s, χ).

As s → 1⁺, the principal character contributes log L(s, χ₀) ~ log(1/(s−1)) → +∞ (it inherits the zeta pole), while the crux is that L(1, χ) ≠ 0 for every non-principal χ. That non-vanishing — the technical heart, hardest for real characters, where Dirichlet used the class number formula — makes the other terms bounded. Hence the sum over primes p ≡ a (mod n) diverges, proving infinitely many primes in every progression a + kn with gcd(a,n)=1. Beyond this, Dirichlet L-functions underpin the generalized Riemann hypothesis, the prime number theorem for progressions, Chebotarev density, and modern analytic number theory.

Three Dirichlet characters modulo 5. Here (ℤ/5ℤ)ˣ = {1,2,3,4} is cyclic of order 4, generated by 2 (since 2¹=2, 2²=4, 2³=3, 2⁴=1). We set i = √−1. Values at multiples of 5 are 0 for every character.

a mod 5	χ₀ (principal)	χ₁ (order 4)	χ₂ (order 2, quadratic)
1	1	1	1
2 (generator g)	1	i	−1
3 = g³	1	−i	−1
4 = g²	1	−1	1
0 (i.e. 5 | a)	0	0	0
sum over period	4	0	0