Counting invertible sums of squares modulo \(n\) and a new generalization of Euler's totient function (Q2834177)

The paper evaluates the function NEWLINE\[CARRIAGE_RETURNNEWLINE \Phi_k(n)=\{(x_1,\ldots,x_k)\pmod n: \gcd(x_1^2+\cdots+x_k^2,n)=1\}. CARRIAGE_RETURNNEWLINE\]NEWLINE Clearly, \(\Phi_1(n)=\varphi(n)\), the more familiar Euler function. It is shown that \(\Phi_k(n)\) is multiplicative for all \(k\geq 1\). As a byproduct the authors also evaluate the number of solutions to NEWLINE\[CARRIAGE_RETURNNEWLINE x_1^2+\cdots+x_k^2\equiv \lambda \pmod n, CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\lambda\) is any fixed residue class coprime to \(n\). The authors also compute the average value of \(\Phi_k(n)\) on the interval \([1,x]\). The arguments are elementary but clever and the paper is well-written. The paper concludes with a couple of problems which should spur further work.