On higher-power moments of $\Delta_{(1)}(x)$

Abstract

Let $d_{(1)}(n)$ be the n-th coefficient of the Dirichlet series $(\zeta '(s))^{2}=\sum _{n=1}^{\infty }d_{(1)}(n)n^{-s}$ in $\mathfrak {R}s>1$, and $\Delta _{(1)}(x)$ be the error term of the sum $\sum _{n\le x}d_{(1)}(n)$. In this paper, we study the higher power moments of $\Delta _{(1)}(x)$ and derive asymptotic formulas for

$$\int _{1}^{T}\Delta _{(1)}^{k}(x)\, dx , \quad k=3,\ldots , 9.$$