Let k, l be non-negative integers and \(\zeta ^{(k)}(s)\) denote the kth derivative of the Riemann zeta function \(\zeta (s).\) Further let \(d_{(k,l)}(n)\) be the nth coefficient of the Dirichlet series \(\zeta ^{(k)}(s)\zeta ^{(l)}(s)=\sum _{n=1}^{\infty }\frac{d_{(k,l)}(n)}{n^{s}}\) for \(\mathfrak {R}s>1,\) and \(\Delta _{(k,l)}(x)\) be the error term of \(\sum _{n\le x}d_{(k,l)}(n).\) In this paper, we will study some properties of \(\Delta _{(k,l)}(x)\), including its “truncated Voronoï formula” , the mean square formula and the higher-power moments.

令k , l为非负整数且\(\zeta ^{(k)}(s)\)表示黎曼 zeta 函数\(\zeta (s).\) 的第k个导数进一步令\(d_ {(k,l)}(n)\)是狄利克雷级数的第n个系数\(\zeta ^{(k)}(s)\zeta ^{(l)}(s)=\sum _{ n=1}^{\infty }\frac{d_{(k,l)}(n)}{n^{s}}\)对于\(\mathfrak {R}s>1,\)和\( \Delta _{(k,l)}(x)\)是\(\sum _{n\le x}d_{(k,l)}(n).\)在本文中，我们将研究\(\Delta _{(k,l)}(x)\) 的一些性质，包括其“截断的 Voronoï 公式”、均方公式和高次幂矩。