On the distribution of powerful numbers
Let (1) \(n = p_1^{\alpha_1}\ldots p_r^{\alpha_r} (p_1 < \ldots < p_r)\) be the canonical decomposition of \(n\). If \(k,\ell \geq 2\) are given integers, then \(n\) is \(k\)-full (or powerful of type \(k\)) if \(n=1\) or if in (1) \(\alpha_j \geq k (j = 1,\ldots,r)\), and \(n\) is \(\ell\)-free if \(n=1\) or if in (1) \(\alpha_j \leq \ell-1 (j = 1,\ldots,r)\). Let \(\Delta_k(x)\) denote the error term in the asymptotic formula for the number of \(k\)-full integers not exceeding \(x\), and let \(N_{k,\ell}(x) (1 < k < \ell)\) denote the number of integers not exceeding \(x\) which are simultaneously \(k\)-full and \(\ell\)-free. The author is interested in upper bounds for \(\Delta_k(x)\) and \(\Delta_{k,\ell}(x,h)\), which represents the error term in the expression for \(N_{k,\ell}(x+h) - N_{k,\ell}(x)\) with suitable \(h>0\). The function \(\Delta_k(x)\) has been extensively studied in the literature, and some of the results are presented in the introductory part of the paper. The author proves five theorems, based on ten lemmas involving estimates in the general divisor problem (the generating function being \(\zeta(a_1s)\ldots\zeta(a_rs)\) for given integers \(1\leq a_1\leq\ldots\leq a_r\)). Four of the theorems involve upper bounds for \(\Delta_{k,\ell}(x,h)\) when \(k = 2,3,4,5\). On the other hand Theorem 1 gives the bounds \[ \Delta_5(x) \ll x^{54/605}\log^8x,\quad \Delta_6(x) \ll x^{67/843}\log^9x, \quad \Delta_7(x) \ll x^{1/14}\log^{10}x. \] Previously \textit{E. Krätzel} [Math. Nachr. 114, 97-104 (1983; Zbl 0545.10030)] obtained \(\Delta_5(x) \ll x^{6/65+\varepsilon}\) and \(\Delta_6(x) \ll x^{13/162+\varepsilon}\), while \textit{P. Shiu} and \textit{A. Ivić} [Ill. J. Math. 26, 576-590 (1982; Zbl 0484.10024)] proved that \(\Delta_7(x) \ll x^{1/14+\varepsilon}\).
scientific article; zbMATH DE number 1114175
scientific article