Higher order divisor functions over values of mixed powers

Abstract: Let $\tau_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$ \sum_{1\leq n_1,n_2, \dots, n_{\ell}\leq X^{\frac{1}{r}} \atop 1\leq n_{\ell+1}\le X^{\frac{1}{s}}}\tau_k(n_1^r+n_2^r+\dots +n_{\ell}^r+n_{\ell+1}^s), $$ where $k\geq 4$, $r\geq 2$, $s\geq 2$ and $\ell\geq 2$ are integers. Previously only special cases are studied.