I have trouble understanding some general concepts related to characters in analytic number theory. In particular, I understand that the $\chi$ function is a completely multiplicative and periodic function from $\mathbb{Z}\rightarrow\mathbb{C}$, and that it plays an important role in the study of Zeta functions and Dirichlet L-functions.
However, I struggle understanding the more general concept of "characters". For instance, I have read the following definition of a character: "the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix" (Wikipedia). This is highly confusing as I have not encountered any matrices in number theory for which the trace was of particular interest. Moreover, I don't see how a "group representation" applies to functions such as the Dirichlet L-function.
So my questions are as follows:
- How is $\chi$ related to matrices or group representations?
- Why are they called "characters"? Why is it not enough to call them "periodic group operators" or something similar? I don't see where the word "character" comes from.
A matrix representation of a group $G$ can be $n$-by-$n$ for any positive integer $n$. In particular it can be $1$-by-$1$. These are not very interesting matrices! It seems idle to maintain a distinction between the matrix $\pmatrix{a}$ and the number $a$, which just happens to be the trace of the matrix $\pmatrix{a}$. So a representation by $1$-by-$1$ complex matrices is really just s group homomorphism $\chi: G\to \newcommand{\C}{\Bbb C}\C^\times$.
If $G$ is finite, then $\chi$ takes values in the group of roots of unity in $\C^\times$. Some author call a group homomorphism $\chi:G\to\C^\times$ a character of $G$. Others call this a quasi-character reserving character for the case where $|\chi(g)|=1$ for all $G$. This distinction is irrelevant when $G$ is finite.
A Dirichlet character in effect is a character on a multiplicative group $\newcommand{\Z}{\Bbb Z}(\Z/N\Z)^\times$. One extends this to a map on $\Z/N\Z$ by taking $\chi(a)=0$ when $\gcd(a,N)>1$. One can then regard this as an $N$-periodic map on $\Z$.
By Class Field Theory, Dirichlet characters correspond to $1$-dimensional representations of the absolute Galois group of $\Bbb Q$. For higher dimensional representations, the analogue of the Dirichlet L-function is the Artin L-function.