Distribution of values of general Euler totient function

Abstract

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 Caldéron et. al. Motivated by the classical works of Schoenberg, Erdős, 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 ${\Phi }_k(n)$.

24 pages