Note on a sum involving the Euler function

TL;DR

This paper establishes an asymptotic upper bound for a sum involving the Euler totient function and the integer part of x/n, contributing to understanding the behavior of such sums as x grows large.

Contribution

The paper provides a new asymptotic upper bound for a sum involving the Euler totient function and the integer part of x/n, advancing knowledge on related number-theoretic sums.

Findings

Proved an asymptotic upper bound for the sum involving ([x/n])

Connected the sum's growth to x log x with explicit constants

Enhanced understanding of sums involving the Euler totient function

Abstract

We prove that $$ \sum_{n \leq x} \varphi([x/n])\leq\bigg(\frac{1380}{4009}+\frac{2629}{4009}\cdot\frac1{\zeta(2)}+o(1)\bigg)x\log x $$ as $x\to\infty$, where $\varphi$ denotes the Euler totient function and $[x]$ denotes the integer part of $x$.