# Note on sums involving the Euler function

## TL;DR

This paper refines estimates for sums involving the Euler totient function and its ratios, improving understanding of their asymptotic behavior for large x, building on recent work by Bordellès et al. and Wu.

## Contribution

It provides more precise asymptotic estimates for sums involving the Euler totient function and its ratios, extending recent research in the area.

## Key findings

- Refined asymptotic estimates for sums involving vy totient function
- Improved bounds on sums vy function ratios
- Enhanced understanding of the sums' growth behavior

## Abstract

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordell\`es et al. and Wu.