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

Jean-Christophe Pain

TL;DR

The article develops explicit finite-sum representations for Euler's totient function $\varphi(n)$ and the divisor-counting function $\tau(n)$ by combining floor-sum identities for gcd, the Menon relation, and the Pillai arithmetic function. It derives two main floor-based expressions for $\varphi(n)$, and a third relation linking $\varphi(n)$ and $\tau(n)$ via $\sum_{\gcd(k,n)=1}\gcd(k-1,n)$; it then expresses $\tau(n)$ in several ways using gcd-sum sums and floor sums, all anchored in the Pillai function framework. These formulas offer finite-sum tools potentially useful for bounding the usual arithmetic functions and for exploring extensions to related functions such as Jordan totients and the Mertens function. The work suggests pathways to further generalizations and applications in analytic and combinatorial number theory. $\varphi(n)$ and $\tau(n)$ thus gain new finite-sum representations tied to gcd-floor structures and classic identities.

Abstract

In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $τ(n)$ 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.

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

TL;DR

The article develops explicit finite-sum representations for Euler's totient function and the divisor-counting function by combining floor-sum identities for gcd, the Menon relation, and the Pillai arithmetic function. It derives two main floor-based expressions for , and a third relation linking and via ; it then expresses in several ways using gcd-sum sums and floor sums, all anchored in the Pillai function framework. These formulas offer finite-sum tools potentially useful for bounding the usual arithmetic functions and for exploring extensions to related functions such as Jordan totients and the Mertens function. The work suggests pathways to further generalizations and applications in analytic and combinatorial number theory. and thus gain new finite-sum representations tied to gcd-floor structures and classic identities.

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.
Paper Structure (6 sections, 2 theorems, 53 equations)

This paper contains 6 sections, 2 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Expression of the greatest common divisor using integer parts
  3. Two expressions of the Euler totient function
  4. Third relation: expression of the Euler totient function from the Menon relation
  5. Relations for the number of divisors using the Pillai function
  6. Conclusion

Key Result

Theorem 1

Let $a$, $b$, be nonnegative real numbers and $f: [a,b]\rightarrow [c,d]$ a bijective increasing function. One has where $k$ is integer, $n(\mathscr{G}_f)$ is the number of points with nonnegative integer coordinates on the graph of $f$ ang the function $g: \mathbb{R}\rightarrow\mathbb{Z}$ is defined by

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof