Note on a sum involving the Euler function

Abstract

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

This paper has been combined with arXiv:1808.00188