On the distribution of square-full and cube-full numbers
An integer \(n\) is said to be \(k\)-full if \(p^k\) divides \(n\) for every prime divisor \(p\) of \(n\). Much has been written on the distribution of such numbers which was first studied by P. Erdös and G. Szekeres in 1935. In 1984 the reviewer [Glasg. Math. J. 25, 127-134 (1984; Zbl 0526.10035)] initiated the short interval problem, which can be transformed to the study of a certain exponential sum in an interval. The author [Lattice points (1988; Zbl 0675.10031)], \textit{D. R. Heath-Brown} [Math. Proc. Camb. Philos. Soc. 110, 1-3 (1991; Zbl 0735.11043)], \textit{H. Q. Liu} [Acta Arith. 64, 129-149 (1993; Zbl 0778.11050)] and the reviewer [Math. Proc. Camb. Philos. Soc. 112, 1-5 (1992; Zbl 0769.11032)] have made recent significant progress in this problem when \(k = 2,3\). An \(\ell\)-free number is one which is not divisible by \(p^\ell\) for any prime \(p\), and in this paper the author considers those \(k\)-full numbers which are also \(\ell\)-free, with \(\ell > k + 1\). He first uses Dirichlet series to derive asymptotic formulae for the number of such numbers up to \(x\). He then studies the short interval problem for \(k = 2,3\), giving significant results which are too complicated to be reported in full here.
scientific article; zbMATH DE number 819238
scientific article