Dirichlet characters

Section 7.4 Dirichlet characters

From this point on, we will not deal with characters on arbitrary finite abelian groups any more, only the groups \((\bbZ/q\bbZ)^\times\text{.}\) In order to lighten the notation \(\chi \in \widehat{(\bbZ/q\bbZ)^\times}\) (especially occurring as a subscript of a summation!), we will just write \(\chi \mod q\text{.}\) Thus, for example, for any \(g \in (\bbZ/q\bbZ)^\times\text{,}\)

\begin{equation*} \sum_{\chi \mod q} \chi(g) = \begin{cases} |G|, \amp \text{if } g = 1, \\ 0, \amp \text{otherwise} \end{cases}\text{.} \end{equation*}

We can say \(\chi\) is a character modulo \(q\) (or mod \(q\)) if \(\chi\) is a character on \((\bbZ/q\bbZ)^\times\text{.}\)

Definition 7.4.1.

Given a character \(f\) modulo \(q\text{,}\) we define a function \(\chi : \bbZ_{\gt 0} \to \bbC\) by

\begin{equation} \chi(a) = \begin{cases} f(a \bmod q), \amp \text{if } (a,q) = 1, \\ 0 \amp \text{otherwise} \end{cases}\text{.}\tag{7.4.1} \end{equation}

This is called a Dirichlet character.

Theorem 7.4.2.

There are exactly \(\phi(q)\) Dirichlet characters modulo \(q\text{.}\) A Dirichlet character \(\chi\) modulo \(q\) has these properties:

\(\chi\) is periodic with period \(q\text{,}\) that is, \(\chi(n+q) = f(n)\) for all \(n\text{.}\)
\(\chi(n)=0\) if and only if \((n,q) \gt 1\text{.}\)
\(\chi\) is completely multiplicative, that is, \(\chi(mn) = \chi(m)\chi(n)\) for all \(m,n\) (including if \((m,n) \gt 1\)).

Conversely, any function \(\chi : \bbN \to \bbC\) satisfying the above properties must be a Dirichlet character modulo \(q\text{.}\)

Proof.

Exercise.

Dirichlet characters inherit some of the properties of group characters, including the following versions of the orthogonality relations.

Theorem 7.4.3.

Let \(q\) be a positive integer and let \(\chi,\psi\) be Dirichlet characters modulo \(q\text{.}\) Then

\begin{equation} \sum_{a \bmod q} \psi(a)\overline{\chi}(a) = \begin{cases} \phi(q), \amp \text{if } \psi = \chi, \\ 0, \amp \text{otherwise} \end{cases}\tag{7.4.2} \end{equation}

where \(a\) runs over a complete residue system modulo \(q\text{.}\)

Proof.

Exercise.

Remark 7.4.4.

Recall that a complete residue system modulo \(q\) is a set of integers with exactly one element in each residue class modulo \(q\text{.}\) Examples of complete residue systems include \(\{0,1,\dotsc,q-1\}\text{,}\) \(\{1,2,\dotsc,q\}\text{,}\) or more generally any set of \(q\) consecutive integers; but also non-consecutive sets, for example \(\{0,1,2,4,8\}\) is a complete residue system modulo \(5\text{.}\)

Theorem 7.4.5.

Let \(a\text{,}\) \(b\) and \(q\) be positive integers such that \((a,q)=1\text{.}\) Then

\begin{equation} \sum_{\chi \bmod q} \chi(a)\overline{\chi}(b) = \begin{cases} \phi(q), \amp \text{if } a \equiv b \pmod{q}, \\ 0, \amp \text{otherwise} \end{cases}\tag{7.4.3} \end{equation}

where \(\chi\) runs over all Dirichlet characters modulo \(q\text{.}\)

Proof.

If \((a,q) \gt 1\) or \((b,q) \gt 1\) then \(\chi(a)\overline{\chi}(b) = 0\) for all \(\chi\text{.}\) Otherwise, we reduce to group characters and the result follows from the orthogonality of group characters.

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