Sums of the higher divisor function of diagonal homogeneous forms in short intervals

Abstract

<abstract><p>Let $ d_k(n) $ denote the k-th divisor function. In this paper, we give the asymptotic formula of the sum</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits_{\substack{x-y\le n_i^r \le x+y \\ i = 1,2,\ldots,l} }{d_k(n_1^r+n_2^r+\ldots+n_l^r)}, $\end{document} </tex-math></disp-formula></p> <p>where $ n_1, n_2, \ldots, n_l \in \mathbb{Z}^+ $, $ k\ge2 $, $ r\ge 3 $ and $ l &gt; 2^{r-1} $ are integers.</p></abstract>