Absolute value of Jacobi sum $J(\chi,\psi)$ for Dirichlet characters

Question

We know that a character $\chi$ on a finite field $\mathbb{F}_q$, $q$ being a power of a prime, is a group homomomorphism $\chi:\mathbb{F}_q^*\rightarrow \mathbb{C}$. For characters $\chi,\psi$ on $\mathbb{F}_q$, the Jacobi sum $J(\chi,\psi)$ has absolute value $\sqrt{q}$. My question is, is there any corresponding result for Dirichlet characters $\chi$ modulo $n$ (primitive or non-primitive) where $n$ is a positive integer or in the special case where $n$ is a power of a prime?

Answer 1

• First note that there is some kind of multiplicativity: write $\chi=\prod_{p^k\| n} \chi_{p^k}$ where $\chi_{p^k}$ is a Dirichlet character modulo $\chi_{p^k}$, you'll get that $$J(\chi,\psi)=\prod_{p^k\| n} J(\chi_{p^k},\psi_{p^k})$$

• Then assume that $\chi,\psi,\chi\psi$ are primitive. It is well-known that $$\chi(a)= \frac{G(\chi)}{n}\sum_{k\bmod n} e^{-2i\pi ak/n}\overline{\chi}(k), \qquad |G(\chi)|^2=n$$ from which $$J(\chi,\psi)= \sum_{a\bmod n} \chi(a)\psi(1-a)$$ $$=\frac{G(\chi)G(\psi)}{n^2}\sum_{a\bmod n}\sum_{k\bmod n}\sum_{l\bmod n} e^{-2i\pi (ak+(1-a)l)/n}\overline{\chi}(k)\overline{\psi}(l) $$ $$ = \frac{G(\chi)G(\psi)}{n^2}\sum_{k\bmod n}n e^{-2i\pi k/n}\overline{\chi}(k)\overline{\psi}(k)=\frac{G(\chi)G(\psi)}{n}\overline{G(\chi\psi)}=\frac{G(\chi)G(\psi)}{G(\chi\psi)}$$ $$|J(\chi,\psi)|=\sqrt{n}$$