On a trigonometric sum and its application (Q580420)

The problem of distribution of square-free numbers in short intervals stands in a close connection with the problem of estimating the number of lattice points \(D(1,2;x)\) in the domain \(\xi \eta^2\leq x\), \(\xi >0\), \(\eta >0\). It is well-known that \[ D(1,2;x)=\zeta (2)x+\zeta (1/2)\sqrt{x}+\Delta (1,2;x), \] \[ \Delta (1,2;x)=\sum_{n^3\leq x}\{\psi (x/n^2)+\psi (\sqrt{x/n})\}+O(1) = O(x^{2/9} \log x), \] which was proved by \textit{H.-E. Richert} [Math. Z. 56, 21--32 (1952; Zbl 0046.25002)]. \textit{P. G. Schmidt} [Abschätzungen bei unsymmetrischen Gitterpunktproblemen (Diss. Göttingen, 1964)] found the improvement \(\Delta (1,2;x)\ll x^{0,2215\ldots}\). The most difficult problem is the estimation of the first sum of \(\Delta(1,2;x)\). The authors now apply the method of exponent pairs for double exponential sums, developed by \textit{B. R. Srinivasan} [Math. Ann. 160, 280--311 (1965; Zbl 0141.04603)], and prove the estimation \[ \sum_{n^3\leq x}\psi(x/n^2) \ll x^{0,21829}. \]