On an error term related to the greatest divisor of \(n\), which is prime to \(k\)
For a fixed positive squarefree integer \(k\) let \(\delta_ k(n)\) denote the greatest divisor of \(n\) which is prime to \(k\) and define the error term \(E_ k\) by \[ \sum_{n\leq x}\delta_ k(n)=(k/2 \sigma (k))x^ 2+E_ k(x). \] \textit{V. S. Joshi} and \textit{A. M. Vaidya} [Topics in Classical Number Theory, Colloq. Budapest 1981, Vol. 1, Colloq. Math. Soc. Janos Bolyai 34, 791--806 (1984; Zbl 0546.10038)] obtained \(E_ k(x)=O(x)\) and for a prime \(p\) they showed \(E_ p(x)=\Omega_{\pm}(x)\). At the conference on Number Theory in Oberwolfach the reviewer announced a joint paper with \textit{J. Herzog} [Arch. Math. 50, 145--155 (1988; Zbl 0616.10035)] where among other things the same result is shown in the general case by using the Tauberian theorem of Hardy, Littlewood and Karamata. In the paper under review this result is obtained by the Erdős-Shapiro method of averaging over arithmetic progressions. Reviewer's remark: In the above mentioned paper J. Herzog and the reviewer pointed out that the Tauberian argument may be replaced by elementary estimates.
divisors prime to a fixed positive squarefree integer
scientific article; zbMATH DE number 4068998