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arXiv · 2505.08434

Explicit formulas for Euler's totient function and the number of divisors

Jean-Christophe Pain
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Abstract

In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of the formulas are obtained using a relation due to Menon and the connections with the Pillai arithmetic function are outlined. The reported expressions may be useful to derive new bounds for the usual arithmetical functions.

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Jean-Christophe Pain. 2025-05-13. Explicit formulas for Euler's totient function and the number of divisors. https://arxiv.org/abs/2505.08434

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