Number of primitive Dirichlet characters of certain order and of bounded conductor

Question

Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that $$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$ My question is how to find an asymptotic expression for the cardinality of the more specific set $$\{\text{$\chi$ primitive Dirichlet characters}\,:\,\chi^d=1,\,q(\chi)\leq Q\}$$ for a fixed $d\in\mathbb{Z}_{\geq0}$. By Lemma 9.3 in Montgomery and Vaughan's Multiplicative number theory, it suffices to count only those characters $\chi$ in the above set modulo a prime power, but I'm not sure how to proceed from there.

Answer 1

You're in luck, because Theorem 2 in this paper of Finch, Sebah, and myself gives an asymptotic formula for exactly this quantity. The order of magnitude is $Q(\log Q)^{\tau(d)-2}$, where $\tau(d)$ is the number of positive divisors of $d$; the leading constant is complicated but is given explicitly in the paper. (Note that there is a difference between "$\chi^d=1$" and "$\chi$ has order $d$", but the same asymptotic formula holds for the two cases.)