Homomorphism Counts Quadratic Residues

We prove that the ratio $\varrho(n)=ϕ(n)/2^{ω(n)}$ of surjective group to ring homomorphism counts between finite cyclic rings admits three simultaneous interpretations that have not previously been connected. It equals the order of the group of squares in $(\mathbb Z/n \mathbb Z)^*$, the degree $[\