On Quotients of Values of Euler's Function on Factorials

TL;DR

This paper studies the divisibility properties of Euler's totient function evaluated at factorials, identifying the minimal factorial index for divisibility and analyzing its asymptotic behavior as parameters grow large.

Contribution

It introduces a new problem of finding the least positive integer c for which a specific quotient involving Euler's totient function on factorials is integral, and analyzes its asymptotic ratio.

Findings

The ratio c(a,b)/(a+b) approaches a limit as a, b tend to infinity.

For all but a density-zero set, c(a,b) exceeds a+b.

The paper characterizes the divisibility conditions of Euler's totient on factorials.

Abstract

Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function $\phi$ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers $a$ and $b$, the least positive integer $c=c(a,b)$ such that the quotient $\phi(c!)/\phi(a!)\phi(b!)$ is an integer. We derive results on the limit of the ratio $c(a,b)/(a+b)$ as $a$ and $b$ tend to infinity. Furthermore, we show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ with an exception of a set of density zero.