  Let $\Phi_k(n)=|\{ (x_1, x_2, \cdots, x_k)\in
\left(\mathbb{Z}/n\mathbb{Z}\right)^k; \ \gcd(x_1^2+x_2^2+ \cdots+ x_k^2,
n)=1\}|$ be a general totient function introduced first by Cald\'{e}ron et. al.
Motivated by the classical works of Schoenberg, Erd\H{o}s, Bateman and Diamond
on the distribution of $\Phi_1(n)$, we prove results on the joint distribution
of $\Phi_k(n)$ for any $k\ge 1$. Additionally, we also exhibit the extremal
order of $\Phi_k(n)$.
