On a sum involving the Euler function

Abstract

We obtain reasonably tight upper and lower bounds on the sum ${\sum }_{n⩽x}\phi \left(\lfloor x/n\rfloor \right)$, involving the Euler functions $\phi$ and the integer parts $\lfloor x/n\rfloor$ 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