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

On a partial sum related to the Euler totient function