On a sum involving the Euler function

TL;DR

This paper derives tight bounds for a sum involving the Euler totient function and the integer parts of reciprocals, advancing understanding of their combined behavior in number theory.

Contribution

It provides new tight upper and lower bounds on a sum involving the Euler totient function and reciprocals, which was previously not well-understood.

Findings

Established tight bounds for the sum involving  and 

Enhanced understanding of the behavior of the Euler function in reciprocal sums

Contributed to analytical techniques for bounding arithmetic sums

Abstract

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers.