The distribution of 4-full numbers (Q1335541)

The distribution of powerful integers of type 4 is considered. A natural number \(n_ 4>1\) is said to be 4-full if each prime factor of \(n_ 4\) divides it at least to the 4-th power. Let \(Q_ 4(x)\) denote the number of 4-full integers not exceeding \(x\). Then \[ Q_ 4(x)= \sum_{k=4}^ 7 r_ k x^{1/k}+ \Delta_ 4(x) \] with some well-defined constants \(r_ k\). There are many estimates of the remainder \(\Delta_ 4(x)\). The best estimation hitherto was established by \textit{H. Menzer} [Monatsh. Math. 107, 69-75 (1989; Zbl 0679.10035)]: \[ \Delta_ 4(x) \ll x^{35/316+ \varepsilon}, \qquad 35/316= 0,1107\dots\;. \] The author now improves this result to \[ \Delta_ 4(x)\ll x^{3626/35461+ \varepsilon}, \qquad 3626/35461= 0,1022\dots\;. \] The method of proof is based on the method of the reviewer and a combination of van der Corput's \(B\)-step with a result of \textit{E. Fouvry} and \textit{H. Iwaniec} [J. Number Theory 33, 311- 333 (1989; Zbl 0687.10028)] for monomials.