On the distribution of \(\left\{x\over n\right\}\). II (Q1913911)

[For Part I, see J. Shandong, Univ., Nat. Sci. Ed. 29, 280-290 (1994; Zbl 0820.11043).] For given large \(x > 1\), let \(\alpha\) and \(R\) satisfy \(0 \leq \alpha < 1\) and \(1 \leq R < x\). Let \(L(x; R, \alpha)\) denote the number of solutions of the inequality \[ \alpha \leq \{x/n\} < \alpha + (R/n) \text{ and } n \leq x \] where \(\{r\}\) denotes the fractional part of the real number \(r\). The author proves that, for any exponent pair \((\kappa, \lambda)\), if \(R < x^{( \lambda + \kappa)/2 (1 + \kappa)} \log x\) then, uniformly for \(\alpha\), \[ L(x; R, \alpha) \ll \min (x^{(\lambda + \kappa)/2 (1 + \kappa)},\;R^{1/2} x^{1/4}) \exp (\log x/ \log \log x). \] Especially \[ L(x; 1, \alpha) \ll x^{1/4} \exp (\log x/ \log \log x). \tag{*} \] These results improve on previous known ones by the author [loc. cit.] where the bound in the inequality (*) is \(x^{(1/3) - (1/36) + \varepsilon}\). The method used in the proof is mainly the contour integration and rational approximation which is quite different from the exponent sum method applied previously by \textit{W. Wang} [Chin. Sci. Bull. 38, 1596-1600 (1993; Zbl 0792.11021)] and the author [loc. cit.].