Sums related to Euler's totient function

TL;DR

This paper derives an upper bound for sums involving the ratio of integers to their Euler totient, and applies this to estimate how many integers exceed a certain ratio, advancing understanding of totient-related sums.

Contribution

It introduces new bounds for sums of ratios involving Euler's totient function and provides applications to counting integers with large ratios.

Findings

Established an upper bound for the sum of (a_n/φ(a_n))^s

Provided bounds on the count of n with a_n/φ(a_n) exceeding t

Extended understanding of totient function sums and their distributions

Abstract

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some restrictions. As applications, for any $t>0$, we obtain an upper bound for the number of $n\in [1,N]$ such that $a_{n}/ \varphi (a_{n})> t$.