Sum of higher divisor function with prime summands

Abstract

Let l ≽ 2 be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{1 \leqslant {n_1},{n_2},...,{n_1} \leqslant {x^{1/2}}} {{\tau _k}(n_1^2 + n_2^2 + ... + n_1^2),} $$\end{document} where τk (n) represents the kth divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{1 \leqslant {p_1},p2,...,{p_1} \leqslant x} {{\tau _k}({p_1} + {p_2} + ... + {p_l}),} $$\end{document} where p1, p2, …, pl are prime variables.