On the error term $\Delta _{(k,l)}(x)$

Abstract

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.