On Wikipedia they state the following identity for the Gauss sum of an imprimitive character: suppose that $\chi: (\mathbb{Z}/m\mathbb{Z})^\times \rightarrow \mathbb{C}$ is a Dirichlet character with conductor $n$. Suppose $\chi\_0$ is the character of modulus $n$ from which $\chi$ is induced. Then
$$G(\chi) = \mu(m/n)\chi\_0(m/n)G(\chi\_0).$$
I would like to find a standard text I could reference for this fact, but I have not found it in my go-to number theory textbooks. Perhaps the result is usually stated in a different way and that is why I am missing it?



Gauss sum of imprimitive characters
See Theorem 9.10 in Montgomery-Vaughan: Multiplicative number theory I (Cambridge University Press, 2006).


