Let k be a natural number. Let \(\delta_ k\) denote the arithmetical function defined by \(\delta_ k(n)=\max \{d\in {\mathbb{N}} :\) \(d| n\), \((d,k)=1\}\). For the error term \(E_ k(x)\), defined by \(E_ k(x)=\sum_{n\leq x}\delta_ k(n)-kx^ 2/2\sigma (k)\), the authors prove: \[ \limsup_{x\to \infty}\frac{E_ k(x)}{x}\leq \frac{1}{2}(1- \frac{1}{p+1})d(k)\quad and\quad \liminf_{x\to \infty}\frac{E_ k(x)}{x}\geq -\frac{1}{2}(1-\frac{1}{p+1})d(k), \] where p is the smallest prime dividing k and d is the divisor function. This improves a result of J. Herzog and Th. Maxsein [Arch. Math. 50, No.2, 145-155 (1988; Zbl 0616.10035)].