On a sum involving the Euler function

Abstract

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

This version incorporate the improvement of the upper bound of Theorem 2.1 in v.1, given by Lixia Dai and Hao Pan in arXiv:1809.10381. Consecutively, Lixia Dai and Hao Pan are now co-authors