Analysis is smooth and continuous; the primes are jagged and arithmetic. To attack a question like "are there infinitely many primes of the form 4k+3?" with calculus, you first need a way for a continuous object — a series, an integral — to see the residue class a number lands in mod q. That translator is the Dirichlet character: a cleverly chosen multiplicative function \chi:\mathbb{Z}\to\mathbb{C} that assigns each integer a complex number depending only on n \bmod q. Twist a Dirichlet series by one of these, and its analytic behaviour suddenly encodes the distribution of primes in a single progression. Dirichlet invented them in 1837 to prove exactly that theorem, and they have been the backbone of the subject ever since.

This page is about the characters themselves — what they are, how many there are, what values they take, and how to write them all down. The star of the show is a small, completely explicit table of numbers; by the end you will be able to build one from scratch.

The definition

The cleanest way to see where these come from: a Dirichlet character is nothing more than a homomorphism \chi:(\mathbb{Z}/q\mathbb{Z})^{*}\to\mathbb{C}^{*} from the group of invertible residues into the nonzero complex numbers, extended by zero to every integer that shares a factor with q. The three bullet points above are exactly what you get by unrolling that one sentence. "Periodic mod q" is "depends only on the residue"; "completely multiplicative" is "respects multiplication of residues"; and the zeros appear because the non-invertible residues have nowhere sensible to go, so we send them to 0.

Setting m=n=1 in multiplicativity gives \chi(1)=\chi(1)^2, and since \chi(1)\neq 0 we must have \chi(1)=1. Every character sends 1 to 1 — a tiny fact worth keeping in your pocket.

The principal character

The simplest homomorphism sends everything to 1. Extended by zero, that gives the principal character mod q, written \chi_0:

It is the indicator function of "coprime to q", dressed up as a character. Every modulus has exactly one principal character, and it is the identity element of the group of characters we are about to meet. All the other characters mod q are called non-principal, and they are the ones that do real work — a key fact for later is that a non-principal character sums to zero over a complete period, \sum_{n=1}^{q}\chi(n)=0, whereas \chi_0 sums to \varphi(q).

How many are there? The dual group

Here is the structural heart of the subject. The invertible residues (\mathbb{Z}/q\mathbb{Z})^{*} form a finite abelian group of order \varphi(q). The characters of any finite abelian group G themselves form a group under pointwise multiplication — the dual group \widehat{G} — and for a finite abelian group this dual is isomorphic to G itself.

The isomorphism becomes concrete the moment (\mathbb{Z}/q\mathbb{Z})^{*} is cyclic — which happens for q = 1,2,4,p^k,2p^k. Then a single primitive root g generates every invertible residue, and a character is pinned down entirely by the one value \chi(g). That is the trick behind the worked table below.

Character values are roots of unity

Why must \chi(n) always be a root of unity (when it is not zero)? Because the group is finite. Every invertible residue n has some order d\mid\varphi(q), meaning n^{d}\equiv 1\pmod q. Apply \chi:

so \chi(n) is a d-th root of unity, and in particular a \varphi(q)-th root of unity. All non-zero character values therefore live on the unit circle, at the corners of a regular polygon. The picture below shows the fourth roots of unity — precisely the values available to characters mod 5, whose group is cyclic of order 4.

Because the values are roots of unity, |\chi(n)|=1 whenever \gcd(n,q)=1, and the complex conjugate \overline{\chi(n)}=\chi(n)^{-1}=\chi(n^{-1}) is itself a character — the conjugate character \overline{\chi}, the inverse of \chi in the dual group.

Real vs complex characters

A character is real (also called quadratic) if all its values are real — necessarily then in \{-1,0,+1\}, since the only real roots of unity are \pm 1. Equivalently, \chi is real iff \chi=\overline{\chi}, iff \chi^2=\chi_0, iff \chi has order 1 or 2 in the dual group. A character that takes a genuinely non-real value (an i, say) is complex. Complex characters come in conjugate pairs \{\chi,\overline{\chi}\}, since \overline{\chi}\neq\chi exactly when some value is non-real.

The quadratic characters are the ones tied to the Legendre symbol and quadratic reciprocity; the mod-5 real character below is literally \chi(n)=\left(\tfrac{n}{5}\right).

The smallest example: the characters mod 4

Here (\mathbb{Z}/4\mathbb{Z})^{*}=\{1,3\} has order \varphi(4)=2, so there are exactly two characters. The principal \chi_0, and one non-principal character \chi_1 determined by \chi_1(3)=-1 (it must be a square root of \chi_1(3^2)=\chi_1(1)=1, and it can't be +1 or we'd be back to \chi_0):

Note the zeros in the even columns — that is \gcd(n,4)>1 at work. The non-principal \chi_1 is real, and it is the one whose L-function is the Leibniz series 1-\tfrac13+\tfrac15-\tfrac17+\cdots=\tfrac{\pi}{4}, the very fact Dirichlet used to separate the 4k+1 and 4k+3 primes.

Worked example: the full character table mod 5

Now the main event. We build all \varphi(5)=4 characters mod 5 explicitly. The plan has three steps.

Step 1 — find a primitive root. Take g=2. Its powers mod 5 are

which run through all of \{1,2,3,4\}=(\mathbb{Z}/5\mathbb{Z})^{*}, so 2 is a primitive root. This gives each unit a discrete logarithm (its index): \operatorname{ind}(1)=0, \operatorname{ind}(2)=1, \operatorname{ind}(4)=2, \operatorname{ind}(3)=3.

Step 2 — choose where g goes. A character is fixed by \chi(2), and since 2 has order 4, \chi(2) must be a fourth root of unity: one of 1,\,i,\,-1,\,-i. Those four choices give the four characters \chi_0,\chi_1,\chi_2,\chi_3 with \chi_k(2)=i^{k}.

Step 3 — fill in the rest by multiplicativity. Using \chi_k(n)=\chi_k(2)^{\operatorname{ind}(n)}=i^{\,k\cdot\operatorname{ind}(n)}. For example \chi_1(3)=\chi_1(2)^{3}=i^{3}=-i and \chi_1(4)=\chi_1(2)^{2}=i^{2}=-1. Do this for every entry and the whole table drops out:

Everything checks out against the theory. There are exactly 4 rows. Every entry is a fourth root of unity or 0. The first column is all 1's (since \chi(1)=1) and the n=5 column is all 0's (since \gcd(5,5)=5>1). The row \chi_2 is real — it is the quadratic character, with +1 on the quadratic residues \{1,4\} and -1 on the non-residues \{2,3\}. And \chi_1 and \chi_3 are a conjugate pair: \chi_3=\overline{\chi_1}=\chi_1^{-1}. As a group the four rows are cyclic of order 4, generated by \chi_1 (\chi_1^2=\chi_2,\ \chi_1^3=\chi_3,\ \chi_1^4=\chi_0) — a perfect mirror of (\mathbb{Z}/5\mathbb{Z})^{*} being cyclic of order 4.

Conductor: primitive vs imprimitive characters

One subtlety keeps every serious calculation honest. A character mod q can secretly be a character of a smaller modulus d\mid q wearing a disguise. Say \chi mod q actually only depends on n\bmod d for some proper divisor d of q; then it is induced by a character \chi^{*} mod d, and we call \chi imprimitive.

A quick example. Mod 4 the real character \chi_1 (with \chi_1(3)=-1) is primitive: its conductor is genuinely 4. But regard it as living mod 12 by insisting it also vanish on multiples of 3 — that induced character mod 12 is imprimitive, with conductor still 4. The two agree on every n coprime to 12, but they differ on integers coprime to 4 yet divisible by 3 (like 3,9,15,\dots), where the mod-4 version gives \pm1 and the induced mod-12 version gives 0. Primitive characters are the "true" building blocks; imprimitive ones are bookkeeping copies, and their L-functions differ from the primitive original only by a handful of Euler factors.

Where this is heading

Characters are the alphabet; the sentences come next. Attach a character to a Dirichlet series to form a Dirichlet L-function L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=\prod_{p}\frac{1}{1-\chi(p)p^{-s}}, whose Euler product exists precisely because \chi is completely multiplicative. The orthogonality relations for characters are then what let a weighted sum of \log L(s,\chi) over all \chi mod q isolate a single residue class — the engine of Dirichlet's theorem. Master the little table above and the rest of the machinery has somewhere solid to stand.