Section 2.1 Dirichlet Characters

Dirichlet characters are certain arithmetic functions, named after P. G. L. Dirichlet, which were introduced by Dirichlet in his 1837 paper [12] proving the infinitude of primes in arithmetic progressions.

Definition 2.1. Dirichlet Characters.

A Dirichlet character modulo $N\text{,}$ usually denoted by $\chi\text{,}$ is a group homomorphism $\chi:(\Z/N\Z)^{\times}\longrightarrow\C^\times\text{,}$ where $N$ is a positive integer.

The positive integer $N$ is called the modulus of $\chi\text{.}$ Let $N\mid M$ be two positive integers. Then, via the following diagram

\begin{gather*} (\Z/M\Z)^\times\longrightarrow(\Z/N\Z)^\times\longrightarrow\C^\times, \end{gather*}

we can think of a character $\chi$ modulo $N$ as a character modulo $M\text{.}$ Therefore, the modulus of a character is not unique.

Example 2.2.

Let $N=5\text{.}$ There are four distinct characters modulo $5\text{.}$ The following table contains the values of all the 4 Dirichlet characters modulo 5:

Table 2.1. Dirichlet characters modulo 5

	$1$	$2$	$3$	$4$
$\chi_{5,1}$	$1$	$1$	$1$	$1$
$\chi_{5,2}$	$1$	$i$	$-i$	$-1$
$\chi_{5,3}$	$1$	$-i$	$i$	$-1$
$\chi_{5,4}$	$1$	$-1$	$-1$	$1$

Note that $N=5$ is a prime. In general, for a prime $p\text{,}$ the group $(\Z/p\Z)^\times$ is a cyclic group due to the existence of a primitive root modulo $p\text{.}$ Therefore, the Dirichlet characters modulo a prime $p$ are characters of the cyclic group $\Z/(p-1)\Z\text{,}$ which are easy to describe in terms of $(p-1)^{\rm th}$ complex roots of unity.

Example 2.3.

We give an example of a Dirichlet character modulo a composite integer. Take $N=15\text{.}$ Note that $(\Z/15\Z)^\times$ is not cyclic. There are eight characters modulo $15$

Table 2.2. Dirichlet characters modulo 15

	$1$	$2$	$4$	$7$	$8$	$11$	$13$	$14$
$\chi_{15,1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$\chi_{15,2}$	$1$	$-i$	$-1$	$i$	$i$	$-1$	$-i$	$1$
$\chi_{15,4}$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$
$\chi_{15,7}$	$1$	$i$	$-1$	$i$	$-i$	$1$	$-i$	$-1$
$\chi_{15,8}$	$1$	$i$	$-1$	$-i$	$-i$	$-1$	$i$	$1$
$\chi_{15,11}$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
$\chi_{15,13}$	$1$	$-i$	$-1$	$-i$	$i$	$1$	$i$	$-1$
$\chi_{15,14}$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$

Generally, for a finite abelian group $G\text{,}$ we denote the set of characters $\chi:G\longrightarrow\C^\times$ by $\widehat{G}\text{.}$ We can give $\widehat{G}$ a group structure in an obvious way: for any two characters $\chi,\psi\in\widehat{G}\text{,}$ we define $\chi\psi(g)$ to be $\chi(g)\psi(g)$ for all $g\in G\text{.}$ It is well-known that $\widehat{G}$ is noncanonically isomorphic to $G$ [47, Lemma 3.1]. As a result, there are $\#(\Z/n\Z)^\times=\varphi(n)$ Dirichlet characters modulo $n\text{.}$ The function $\varphi$ is also known as Euler’s Totient function.

Another way to look at Dirichlet characters is to consider them as functions over all of $\Z$ via the following recipe: For any Dirichlet character $\chi$ modulo $N\text{,}$ we may think of $\chi$ as a function defined on all of $\Z$ as follows:

\begin{align*} \chi(a)\colonequals\left\{\begin{matrix} \chi(a\pmod{N}) \amp\gcd(a,N)=1\\0 \amp \gcd(a,N)\gt 1 \end{matrix}\right. \end{align*}

for all $a\in\Z\text{.}$ This makes $\chi$ into a completely multiplicative $N$-periodic arithmetic function

An arithmetic function is any function $f:\Z\longrightarrow\C\text{.}$

. For $N\mid M\text{,}$ as noted above, $\chi$ can be also thought of as an $M$-periodic function, which motivates the following definition:

Definition 2.4. Conductor.

For any Dirichlet character $\chi\text{,}$ the smallest positive integer $n=n(\chi)$ such that $\chi$ is $n$-periodic, is called the conductor of $\chi$ and is denoted as $f_\chi\text{.}$

Therefore, the conductor $f_\chi$ of a Dirichlet character $\chi$ is the smallest modulus of $\chi\text{;}$ hence is uniquely determined by $\chi\text{.}$

For several purposes, Dirichlet characters are classified into two categories:

\begin{align*} \text{$\chi$ is called}\left\{\begin{matrix} \text{\textit{even}, if}\amp\chi(-1)=1\\\text{\textit{odd}, if}\amp\chi(-1)=-1.\end{matrix}\right. \end{align*}

Note that since $\#(\Z/N\Z)^\times=\varphi(N)\text{,}$ for any $a\in(\Z/N\Z)^\times\text{,}$ we have $\chi(a)^{\varphi(N)}=\chi(a^{\varphi(N)})=\chi(1)=1$ and hence the values of a Dirichlet character modulo $N$ are $\varphi(N)^{\rm th}$ complex roots of unity. So, a Dirichlet character modulo $N$ is a group homomorphism between the multiplicative groups $(\Z/N\Z)^\times$ and the unit circle $\bbS^1\text{.}$

Subsection 2.1.1 Primitive Dirichlet characters

As noted above, if $N\mid M\text{,}$ then an integer coprime to $M$ is also coprime to $N\text{.}$ So, for an element $a\in (\Z/M\Z)^\times\text{,}$ $a\pmod{N}$ belongs to $(\Z/N\Z)^{\times}\text{.}$ So, the set of integers coprime to $N$ contains the set of integers coprime to $M$ even if $\varphi(N)\leq\varphi(M)\text{.}$ Therefore, from a Dirichlet character $\chi_1$ modulo $N$ we may create another Dirichlet character $\chi_2$ modulo $M$ as follows: define $\chi_2(a)\colonequals\chi_1(a)$ if $a\in(\Z/N\Z)^{\times}$ and $\chi_2(a)=0$ otherwise. Note that we may create infinitely many Dirichlet characters in this way, one for each multiple of $N\text{.}$ While $\chi_1$ and $\chi_2$ are technically two ‘different’ characters, one is modulo $N\text{,}$ and the other is modulo $M\text{,}$ they are ‘essentially’ the same! So, for a Dirichlet character $\chi$ modulo $N\text{,}$ one might ask if there is a divisor $d$ of $N$ and another Dirichlet character $\psi$ modulo $d$ such that $\chi$ is obtained from $\psi$ using the recipe described above, which motivates the following definition:

Definition 2.5. Induced and Primitive Characters.

A Dirichlet character $\psi$ modulo $M$ is induced by a Dirichlet character modulo $N\text{,}$ where $N\mid M\text{,}$ if $\psi(a)=\chi(a)$ whenever $a$ is coprime to $M\text{.}$ If $\chi$ is a Dirichlet character such that no Dirichlet characters except for itself induce $\chi\text{,}$ then $\chi$ is called primitive.

Example 2.6.

Let $\chi:(\Z/8\Z)^\times\longrightarrow\C^\times$ be defined by $\chi(1)=\chi(5)=1$ and $\chi(3)=\chi(7)=-1\text{.}$ Note that $\chi(n+4)=\chi(n)$ for all $n\in(\Z/8\Z)^\times\text{.}$ Therefore, $\chi$ is induced by the character $\psi$ modulo $4$ defined by $\psi(1)=1$ and $\psi(3)=-1\text{.}$ Since $\psi$ is not $2$-periodic, we conclude that $\chi$ is induced by the primitive Dirichlet character $\psi$ modulo $4\text{.}$ Also, $4$ is the minimal period of $\chi$ and hence, $f_\chi=4\text{.}$

To keep it brief, we will state some basic facts about Dirichlet characters without proof. One may refer to any standard text in number theory for detailed proofs, such as [47, Chapter 3] and [31, Chapter 4, 9].

Fact 1. Let $\chi$ be a Dirichlet character modulo $N\text{.}$ The conductor $f_\chi$ of $\chi$ divides $N$ and there is a unique primitive Dirichlet character $\widetilde{\chi}$ modulo $f_\chi$ such that $\widetilde{\chi}$ induces $\chi$ [31, Theorem 9.2].

Fact 2. The conductor $f_\chi$ of $\chi$ is the period of the unique primitive Dirichlet character $\widetilde{\chi}$ associated to $\chi\text{.}$

The following Lemma will be useful later:

Lemma 2.7.

For a positive integer $N\text{,}$ let $\mathcal{X}(N)$ be the set of primitive Dirichlet characters of modulus (as defined in page 5) dividing $N\text{.}$ Then, there is a canonical bijection

\begin{align*} \widehat{(\Z/N\Z)^\times}\amp\xrightarrow{\sim}\mathcal{X}(N)\\ \chi\amp\mapsto\widetilde{\chi}. \end{align*}

Proof.

By [31, Theorem 9.2], the map $\chi\mapsto\widetilde{\chi}$ is injective. For any character $\widetilde\chi\in\mathcal{X}(N)$ modulo $f_\chi\text{,}$ we have $f_\chi\mid N\text{,}$ by definition of $\mathcal{X}(N)\text{.}$ Hence, we may construct $\chi$ modulo $N$ as follows:

\begin{align*} \chi(a) =\left\{\begin{matrix} \widetilde\chi(a) \amp\gcd(a,N)=1\\0\amp\gcd(a,N)\gt 1 \end{matrix}\right. \end{align*}

Clearly, $\chi\in\widehat{(\Z/N\Z)^\times}$ and $\widetilde\chi$ induces $\chi\text{.}$ Therefore, surjectivity follows from the uniqueness of $\widetilde\chi$ as $\chi$ maps to $\widetilde\chi\text{.}$

Lemma 2.7 shows that we may identify any Dirichlet character $\chi$ with the unique primitive Dirichlet character that induces $\chi\text{.}$ In this way, the modulus of $\chi$ equals its conductor $f_\chi\text{.}$

Remark 2.8.

The multiplication of two Dirichlet characters in $\mathcal{X}(N)$ is not necessarily merely the point-wise product. According to fact 3, we may identify any Dirichlet character with its unique associated primitive character and thereby think of each Dirichlet character as primitive modulo its conductor. Let $\chi$ and $\psi$ be any two Dirichlet characters of conductors $f_\chi$ and $f_\psi$ respectively. Then $\chi$ and $\psi$ are elements of $\mathcal{X}(\lcm(f_\chi,f_\psi))\text{.}$ Consider the map

\begin{gather*} \lambda:(\Z/\lcm(f_\chi,f_\psi)\Z)^\times\longrightarrow\C^\times \end{gather*}

given by $a\mapsto\chi(a)\psi(a)\text{.}$ Then $\lambda$ is a character modulo $\lcm(f_\chi,f_\psi)\text{.}$ But the product $\chi\psi$ is defined to be the unique primitive Dirichlet character that induces $\lambda\text{.}$ When $\gcd(f_\chi,f_\psi)=1\text{,}$ by [31, Theorem 9.3], $\chi\psi$ is a primitive Dirichlet character of conductor $f_\chi f_\psi\text{.}$ Viewing every character as a primitive Dirichlet character plays an important role in multiplying two arbitrary Dirichlet characters.

Example 2.9.

Let $\chi:(\Z/12\Z)^\times\longrightarrow\C^\times$ be defined by $\chi(1)=\chi(11)=1$ and $\chi(5)=\chi(7)=-1$ and $\psi:(\Z/3\Z)^\times$ be defined by $\psi(1)=1$ and $\psi(2)=-1\text{.}$ Then the product character $\chi\psi$ modulo $\lcm(12,3)=12$ has values $\chi\psi(1)=1,\chi\psi(5)=\chi(5)\psi(5)=1,\chi\psi(7)=-1\text{,}$ and $\chi\psi(11)=-1\text{.}$ Note that $\chi\psi(n+4)=\chi\psi(n)$ for all $n\in(\Z/12\Z)^\times$ and hence $f_{\chi\psi}=4$ (strictly smaller than $\lcm(f_\chi,f_\psi)$).

The trivial character $\mathds{1}:\Z\longrightarrow\C^\times$ is defined as $\mathds{1}(n)=1$ for all $n\in\Z\text{.}$ The principal character modulo $N\text{,}$ denote as $\mathds{1}_N\text{,}$ is the character induced by $\mathds{1}\text{.}$ One advantage of considering every Dirichlet character $\chi$ as a primitive character modulo $f_\chi$ is that we have only one universal trivial character modulo $1$ and not one for each modulus.

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	\(1\)	\(2\)	\(3\)	\(4\)
\(\chi_{5,1}\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{5,2}\)	\(1\)	\(i\)	\(-i\)	\(-1\)
\(\chi_{5,3}\)	\(1\)	\(-i\)	\(i\)	\(-1\)
\(\chi_{5,4}\)	\(1\)	\(-1\)	\(-1\)	\(1\)

	\(1\)	\(2\)	\(4\)	\(7\)	\(8\)	\(11\)	\(13\)	\(14\)
\(\chi_{15,1}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(\chi_{15,2}\)	\(1\)	\(-i\)	\(-1\)	\(i\)	\(i\)	\(-1\)	\(-i\)	\(1\)
\(\chi_{15,4}\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)	\(1\)
\(\chi_{15,7}\)	\(1\)	\(i\)	\(-1\)	\(i\)	\(-i\)	\(1\)	\(-i\)	\(-1\)
\(\chi_{15,8}\)	\(1\)	\(i\)	\(-1\)	\(-i\)	\(-i\)	\(-1\)	\(i\)	\(1\)
\(\chi_{15,11}\)	\(1\)	\(-1\)	\(1\)	\(1\)	\(-1\)	\(-1\)	\(1\)	\(-1\)
\(\chi_{15,13}\)	\(1\)	\(-i\)	\(-1\)	\(-i\)	\(i\)	\(1\)	\(i\)	\(-1\)
\(\chi_{15,14}\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(1\)	\(-1\)	\(-1\)	\(-1\)