Higher order divisor functions over values of mixed powers

Abstract

Let $\ddot{I}{„}_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$\sum _{\genfrac{}{}{0ex}{}{1\le n_1,n_2,\dots ,n_{\ell }\le X^{\frac{1}{r}}}{1\le n_{\ell +1}\le X^{\frac{1}{s}}}}\ddot{I}{„}_k(n_1^r+n_2^r+\cdots +n_{\ell }^r+n_{\ell +1}^s),$$ where $k\ge 4$, $r\ge 2$, $s\ge 2$ and $\ell \ge 2$ are integers. Previously only special cases are studied.

19 pages