  Walfisz (1963) proved the asymptotic formula \[ \sum_{n\le x}\varphi(n) =
\frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{4}{3}}), \]
which improved the error term estimate of Mertens (1874) and had been the best
possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved
Walfisz's error term estimate to \[ \sum_{n\le x}\varphi(n) =
\frac{3}{\pi^2}x^2+O(x(\log x)^{\frac{2}{3}}(\log\log x)^{\frac{1}{3}}). \] We
generalize Liu's result to a certain class of arithmetic functions and improve
the result of Balakrishnan and P\'etermann (1996). To this end, we provide a
refined version of Vinogradov's combinatorial decomposition available for a
wider class of multiplicative functions.
