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arXiv · 2304.02540

Distribution of values of general Euler totient function

Debika Banerjee·Bittu Chahal·Sneha Chaubey·Khyati Khurana
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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 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 $\Phi_k(n)$.

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Debika Banerjee, Bittu Chahal, Sneha Chaubey, Khyati Khurana. 2023-04-05. Distribution of values of general Euler totient function. https://arxiv.org/abs/2304.02540

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