On the average behaviour of the largest divisor of \(n\) prime to a fixed integer \(k\) (Q1190624)

For a fixed square-free integer \(k\), let \(\delta_ k(n)\) be the largest divisor of \(n\), which is coprime to \(k\) and put \(E_ k(x)=\sum_{n\leq x} \delta_ k(n)-kx^ 2/2\sigma(k)\), where \(\sigma(k)\) is the sum of the divisors of \(k\). In 1981, \textit{V. S. Joshi} and \textit{A. M. Vaidya} [Topics in classical number theory, Colloq. Math. Soc. János Bolyai 34, 791-806 (1984; Zbl 0546.10038)] proved \(E_ k(x)=O(x)\) and \(E_ k(x)=\Omega_ \pm (x)\). Let \(S_ k=\limsup_{x\to\infty} E_ k(x)/x\) and \(I_ k=\liminf_{x\to\infty} E_ k(x)/x\), Joshi and Vaidya [op. cit.] also proved that \(S_ p=p/(p+1)\) and \(I_ k=-p/(p+1)\), where \(p\) is a prime. When the number \(\omega(k)\) of (distinct) prime divisors of \(k\) exceeds 1, however, the exact values of \(S_ k\) and \(I_ k\) are not known. \textit{J. Herzog} and \textit{T. Maxsein} [Arch. Math. 50, 145-155 (1988; Zbl 0616.10035)] proved that (1) \(S_ k\geq k/\sigma(k)\) and \(I_ k\leq- k/\sigma(k)\). \textit{S. D. Adhikari}, \textit{R. Balasubramanian} and \textit{A. Sankaranarayanan} [Indian J. Pure Appl. Math. 19, 830-841 (1988; Zbl 0655.10038)] proved the above results by applying a simpler method. Recently, \textit{S. D. Adhikari} [Arch. Math. 58, 257-264 (1992; Zbl 0725.11047)] gave an upper bound of \(S_ k\) and a lower bound of \(I_ k\). In the paper under review, the author proves that \(I_ k=-S_ k\) and he improves the lower bound for \(S_ k\) in (1). For example, he proves that \[ S_ k\geq {k \over {\sigma(k)}}+{{q-1} \over {3(q+1)}}, \qquad \text{for}\qquad k=2q\geq 6, \] where \(q\) is a prime.