Let $d$ be a positive integer, and suppose $c$ is an integer such that $\gcd(c,d) = 1$. Then the following identity holds:


$$\displaystyle \left \lvert \{b \pmod{d} : b^2 \equiv c \pmod{d} \}\right \rvert = \sum\_{\substack{\chi \pmod{d} \\ \chi^2 = \chi\_0}} \chi(c).$$


What is the correct analogue for the right hand side when we replace the left hand side with


$$\displaystyle \left \lvert \{b \pmod{d} : b^k \equiv c \pmod{d}\}\right \rvert$$


with $k \geq 3$?



Analogues of an identity involving quadratic characters
If $\gcd(c,d) = 1$ we have


$$\displaystyle \left \lvert \{b \pmod{d} : b^k \equiv c \pmod{d} \}\right \rvert = \sum\_{\substack{\chi \pmod{d} \\ \chi^k = \chi\_0}} \chi(c).$$


This is not really anything to do with numbers - it works in any finite abelian group, and here we are applying it to the multiplicative group of invertible residue classes mod $c$.


It's a consequence of orthogonality of characters for the group $G/ G^k$.


