While I have understood the properties of Dirichlet characters and how to construct tables for them, I am incredibly confused by the notation. Here are the notations in use in the book Introduction to Analytic Number Theory (Tom M. Apostol), Theorem 6.14.
Definition Dirichlet characters. Let $G$ be the group of reduced residue classes modulo $k$. Corresponding to each character $f$ of $G$ we define an arithmetical function $\chi = \chi_f$ as follows:
\begin{align*} \chi(n) & = f(\hat{n}) & \text{ if } (n, k) = 1, \\ \chi(n) & = 0 & \text{ if } (n, k) > 1. \end{align*} The function $\chi$ is called a Dirichlet character modulo $k$. The principal character $\chi_1$ is what which has the properties $$ \chi_1(n) = \begin{cases} 1 & \text{ if } (n, k) = 1, \\ 0 & \text{ if } (n, k) > 1. \end{cases} $$
But previously, in section 6.6, the principal character of a group $G$ was denoted by $f_1$ and the others denoted by $f_2$, $f_3$, $\dots$. So if we choose the principal character $f_1$ and find the corresponding principal Dirichlet character, should it not be denoted as $\chi_{f_1}$? Is $\chi_1$ simply a shorthand for $\chi_{f_1}$?
In that case, is this a more accurate way of defining the notation for Dirichlet characters?
- Corresponding to each character $f_i$ of $G$ we define an arithmetical function $\chi_i$ as follows: \begin{align*} \chi_i(n) & = f_i(\hat{n}) & \text{ if } (n, k) = 1, \\ \chi_i(n) & = 0 & \text{ if } (n, k) > 1. \end{align*}
- Then use $\chi_i(n)$ notation everywhere and don't use $\chi(n)$ anywhere? Why is the secondary notation $\chi(n)$ useful?