Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function

Abstract

In this paper we introduce and study a family ${\Phi }_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd (x_1^2+\dots +x_k^2,n)=1$ with $x_1,\dots ,x_k\in \mathbb{Z}/n\mathbb{Z}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over $\mathbb{Z}/n\mathbb{Z}$. We prove that ${\Phi }_k$ is multiplicative for every $k$, we obtain an explicit formula for ${\Phi }_k(n)$ in terms of the prime-power decomposition of $n$ and derive an asymptotic formula for ${\sum }_{n\le x}{\Phi }_k(n)$. As a tool we investigate the multiplicative arithmetic function that counts the number of solutions to $x_1^2+\dots +x_k^2\equiv \lambda$ (mod $n$) for $\lambda$ coprime to $n$, thus extending an old result that dealt only with the prime $n$ case.

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• A generalization of Menon's identity
• Counting solutions of quadratic congruences in several variables revisited
• On the structure of quaternion rings over $\mathbb{Z}/n\mathbb{Z}$
• Fermat test with gaussian base and Gaussian pseudoprimes