On a partial sum related to the Euler totient function

Abstract

Recently, Bordell\'{e}s, Dai, Heyman, Pan and Shparlinski in \cite{Igor} considered a partial sum involving the Euler totient function and the integer parts $\lfloor x/n\rfloor$ function. Among other things, they obtained reasonably tight upper and lower bounds for their sum using the theory of exponent pairs and in particular, using a recently discovered Bourgain's exponent pair. Based on numerical evidences, they also pose a question on the asymptotic behaviour for their sum which we state here as a conjecture. The aim of this paper is to prove an asymptotic formula for the average of their sum in the interval $[1,x]$. We show via Perron's formula that this average is a certain weighted analogue of the sum that Bordell\'{e}s \textit{et al} considered in their paper. Further, we show that their conjecture is true under certain conditions. Our proof involves Perron's contour integral method besides some analytic estimates of the Riemann zeta function $\zeta(s)$ in the zero-free region.