I was wondering if there was any value to generalising the definition of Dirichlet characters to more general rings in both domain and codomain. For example, a Dirichlet character mod $m$ can be defined as a completely multiplicative function $\chi:\mathbb{Z}\to\mathbb{C}$ that is periodic with period $m$ and $\chi(a)\neq0$ if and only if $\gcd(a,m)=1$. In a straightforward generalisation of this is, let $R$ be a ring and $F$ be a field. A Dirichlet character from $R$ to $F$ mod $I$, an ideal of $R$, could then be a function $\chi:R\to F$ such that $\chi(a)\chi(b)=\chi(ab)$, for all $i\in I$, $\chi(a+i)=\chi(a)$, and $\chi(a)\neq0$ if and only if $\langle a\rangle+I=R$. The idea seems very pretty, does it have any uses?
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4$\begingroup$ The case when $R= \mathbb F_p[t]$ is indeed well studied. You get a complete (and simpler) parallel theory of $L$-functions attached to characters, a (proven!) version of the Riemann Hypothesis and so on. I would recommend looking at the first couple of chapters of Rosen's wonderful book "Number Theory in Function Fields". $\endgroup$Arkady– Arkady2022-08-27 22:26:17 +00:00Commented Aug 27, 2022 at 22:26
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$\begingroup$ Your description is correct, but I think there's a better description (even in the classical situation): a multiplicative character from $R$ to $F$ mod $I$ is really a group homomorphism from $(R/I)^\times$ to $F^\times$, where ${}^\times$ denotes the group (under multiplication) of units of the given ring. This homomorphism can be considered a function on $R/I$ by defining it to be $0$ on the non-units of $R/I$, and then lifted to a function on $R$ in the natural way. $\endgroup$Greg Martin– Greg Martin2022-08-27 23:38:03 +00:00Commented Aug 27, 2022 at 23:38
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For $\chi_4$ the non-trivial Dirichlet character $\bmod 4$ we have $\chi_4(-1)=-1$, so it is interesting to consider $$\psi_4(n)=\frac{n}{|n|} \chi_4(n)$$ which is a Hecke character, the point being that it is well-defined on the non-zero ideals of $\Bbb{Z}$.
This generalizes to number fields, for example $$\psi(a) = \frac{a^2}{|a|^2} \phi(a),\qquad\phi(a) = \begin{cases}1 \text{ if } a = \pm 1 \bmod 2+i\\ -1 \text{ if } a= \pm 2 \bmod 2+i\\ 0 \text{ if } a = 0\bmod 2+i\end{cases}$$ is a Hecke character of $\Bbb{Z}[i]$, and we can consider its Hecke L-function $$L(s,\psi)=\frac14 \sum_{a,b\in \Bbb{Z}^2-(0,0)} (a^2+b^2)^{-s} \psi(a+ib)$$ Similarly to Dirichlet L-functions it factorizes into an Euler product, it has a functional equation and analytic continuation, a prime number theorem, a Riemann hypothesis.
