Explicit formulas for Euler's totient function and the number of divisors
Number Theory
2025-05-14 v1
Abstract
In this article, we present relations for the Euler totient function and the number of divisors 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.
Cite
@article{arxiv.2505.08434,
title = {Explicit formulas for Euler's totient function and the number of divisors},
author = {Jean-Christophe Pain},
journal= {arXiv preprint arXiv:2505.08434},
year = {2025}
}