The distribution of square-full integers
A positive integer \(n\) is said to be square-full if \(p\mid n\) implies that \(p^2\mid n\), where \(p\) are prime numbers. Let \(Q(x)\) denote the number of square-full integers not exceeding \(x\), and \(\Delta (x):= Q(x)- (\zeta (3/2)/ \zeta (3)) x^{1/2}- (\zeta (2/3)/ \zeta (2)) x^{1/3}\). Assuming that the Riemann hypothesis is true, several authors considered an upper bound estimate of \(\Delta (x)\). In the present paper, the author proves the following theorem: Assuming the Riemann hypothesis, then \(\Delta (x)= O(x^{9/ 58+ \varepsilon})\) for any \(\varepsilon>0\). Only recently, \textit{X. D. Cao} [Period. Math. Hung. 28, 43-54 (1994; Zbl 0803.11047)] obtained \(\Delta (x)= O(x^{5/ 33+ \varepsilon})\) under the same condition. \{Reviewer's remark: In 1995, in his thesis, \textit{Y. C. Cai} has established that, assuming the Riemann hypothesis, \(\Delta (x)= O(x^{4/ 27+ \varepsilon})\)\}.
scientific article; zbMATH DE number 750858