On the distribution of $\phi(\sigma(n))$

TL;DR

This paper derives explicit upper bounds on the count of positive integers up to x for which the Euler totient of the sum of divisors exceeds a constant multiple of n, refining previous results by Alaoglu and Erdős.

Contribution

It provides explicit upper bounds on the distribution of integers where f(((n))) exceeds a given multiple of n, improving earlier asymptotic estimates.

Findings

Established explicit upper bounds for the count of such integers.

Refined previous asymptotic results by Alaoglu and Erds.

Enhanced understanding of the distribution of f(((n))) relative to n.

Abstract

Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This is a refinement of a result of Alaoglu and Erd\H{o}s.