Distribution of values of general Euler totient function

Let ${\mathrm{\Phi }}_k(n)=\mathrm{\mid }\{(x_1,x_2,\cdots ,x_k)\in {\left(\mathbb{Z}\mathrm{/}n\mathbb{Z}\right)}^k;\gcd (x_1^2+x_2^2+\cdots +x_k^2,n)=1\}\mathrm{\mid }$ be a general totient function introduced first by Caldéron et. al. Motivated by the classical works of Schoenberg, Erdős, Bateman and Diamond on the distribution of ${\mathrm{\Phi }}_1(n)$, we prove results on the joint distribution of ${\mathrm{\Phi }}_k(n)$ for any $k\ge 1$. Additionally, we also exhibit the extremal order of ${\mathrm{\Phi }}_k(n)$.