On the Sum of Divisors of Mixed Powers

Let $d(n)$ denote the Dirichlet divisor function. Define \begin{equation*} $\mathcal{S}${k}(x)=\sum{\substack{1\leqslant n_1,n_2,n_3 \leqslant x^{1/2} \ 1\leqslant n_4\leqslant x^{1/k} }} d(n_1^2+n_2^2+n_3^2+n_4^k), \qquad 3\leqslant k\in $\mathbb{N}$. \end{equation*} In this paper, we establish an asymptotic formula of ${\mathcal{S}}_k(x)$ and prove that \begin{equation*} $\mathcal{S}$_k(x)=C_1(k)x^{3/2+1/k}\log x+C_2(k)x^{3/2+1/k}+O(x^{3/2+1/k-\delta_k+\varepsilon}), \end{equation*} where $C_1(k),C_2(k)$ are two constants depending only on $k,$ with $\delta_3=\frac{19}{60},,\delta_4=\frac{5}{24},,\delta_5=\frac{19}{140},,\delta_6=\frac{25}{192},, \delta_7=\frac{457}{4032},,\delta_k=\frac{1}{k+2}+\frac{1}{2k^2(k-1)}$for$k\geqslant8.$