On an error term involving the totient function (Q1123226)

Let \(\phi\) (n) denote Euler's totient function. \textit{R. Sitaramachandrarao} [Rocky Mt. J. Math. 15, 579-588 (1985; Zbl 0584.10027)] proved the asymptotic formula \[ \sum_{n\leq x}(n/\phi (n))=315 \zeta (3)/(2\pi^ 4)x-(\log x)/2-c_ 1+E(x) \] where \(c_ 1\) is a positive constant, \(E(x)\ll (\log x)^{2/3}\) and \(\int^{x}_{0}E(t) dt\ll x^{4/5}\). The present author obtains the following mean square result for the error term E(x): \[ \int^{x}_{0}E^ 2(t) dt=Cx+O(x^{4/5}(\log x)^{3/5}(\log \log x)^{6/5}),\quad C>0. \] The main ingredients of his proof are Sitaramachandrarao's representation \[ E(t)=-\sum_{n\leq y(t)}(\mu^ 2(n)/\phi (n))B_ 1(t/n)+O(t^{-1/5}(\log t)^{3/5}(\log \log t)^{6/5}) \] where \(y(t)=t^{4/5}(\log t)^{-2/5}(\log \log t)^{- 4/5})\) and \(B_ 1(\xi)=\xi -[\xi]-1/2\) as well as some estimates due to \textit{A. Walfisz} [Math. Z. 34, 448-472 (1931; Zbl 0003.10301)] such as \[ \sum_{m,n\leq y}\sum_{u,v\geq 1,\quad um\neq vn}(uv)^{-1} | um- vn|^{-1}\quad \ll \quad y \log y. \]