On Higher-Power Moments of ${\mathrm{\Delta }}_a(x)$ for $-1\mathrm{/}2<a<0$

Let $-1\mathrm{/}2<a<0$ be a fixed real number and \begin{equation*} Δ_{a}(x)=\sideset{}{'}\sum_{n\leq x} σ_a(n)-ζ(1-a)x-\frac{ζ(1+a)}{1+a}x^{1+a}+\frac{1}{2}ζ(-a). \end{equation*} In this paper, we investigate the higher–power moments of ${\mathrm{\Delta }}_a(x)$ and give the corresponding asymptotic formula for the integral ${\int }_1^T{\mathrm{\Delta }}_a^k(x)\mathrm{d}x$, which constitutes an improvement upon the previous result of Zhai [9] for $k=3,4,5$ and an enlargement of the upper bound of $k$ to $7$.