On an error term of Chowla. I--III
\textit{S. Chowla} [J. Indian Math. Soc. 18, 138--141 (1929; JFM 55.0106.03)] showed that \(\sum_{n\leq x} \bigl(\frac{\varphi(n)}{n}\bigr)^r=\alpha(r)x+E(x;r)\), with \(E(x;r)\ll \log^rx\), for any positive integer \(r\). Now let \(r\) be an arbitrary positive real number. It is shown firstly that \[ \int^x_1 E(t;r)\,dt\ll x\varepsilon(x;r), \] where \[ \varepsilon(x;r)=\begin{cases} \text{exp}\{-A(\log x)^{3/5}(\log\log x)^{-1/5}\}& \quad r=1,\\ \text{exp}\{-A(\log x)^{1/3}(\log\log x)^{1/3}\}& \quad r=2,3,\dots\\ (\log x)^{-r} & \quad r\notin\mathbb N,\end{cases} \] (\(A\) being a positive constant). There is a corresponding result for \(\sum_{n\leq x} E(n;r)\). The case \(r=1\) is due to \textit{D. Suryanarayana} and \textit{R. Sitaramachandrarao} [Ark. Mat. 10, 99--106 (1972; Zbl 0233.10026)]. It is also shown that \[ \int^x_1 E^2(t;r)\,dt/t\sim c(r)\log x, \] for some constant \(c(r)>0\), generalizing results for \(r=1\) due to \textit{S. Chowla} [Math. Z. 35, 279--299 (1932; Zbl 0004.10202)] and \textit{R. Bellman} [Analytic number theory: An introduction, Benjamin Cummings, Redwood City, CA (1980; Zbl 0448.10001)]. Finally it is shown that \[ E(x;r)=\Omega_\pm((\log\log x)^{1/2}), \] generalizing a result of \textit{H. Montgomery} [Proc. Indian Acad. Sci., Math. Sci. 97, 239--245 (1983; Zbl 0656.10042)].
scientific article; zbMATH DE number 1017563