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

TL;DR

This paper derives explicit formulas for Euler's totient function and the divisor count using finite sums and gcd relations, potentially aiding in establishing new bounds for these functions.

Contribution

It introduces novel explicit formulas for er and au(n) based on finite sums and gcd relations, expanding analytical tools for number theory.

Findings

Formulas relate er and au(n) to finite sums of integer parts and gcds.

Connections with Menon's relation and Pillai's function are established.

Potential applications include deriving new bounds for arithmetical functions.

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.