Quelques in\'egalit\'es effectives entre des fonctions arithm\'etiques usuelles

TL;DR

This paper establishes effective upper bounds relating the ratio of n to Euler's totient function and the sum of divisors to other divisor functions, providing insights into inequalities among common arithmetic functions.

Contribution

It introduces new effective bounds for ratios involving n, Euler's totient, sum of divisors, and divisor count functions, enhancing understanding of their interrelations.

Findings

Derived upper bounds for n/φ(n) in terms of φ(n)

Established bounds for σ(n)/n based on τ(n)

Improved inequalities among divisor functions

Abstract

Let us denote by $\tau(n)$ and $\si(n)$ the number and the sum of the divisors of $n$ and by $\vfi$ Euler's function. We give effective upper bounds for $\frac{n}{\vfi(n)}$ in terms of $\vfi(n)$, and for $\frac{\si(n)}{n}$ in terms of $\tau(n)$.