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-δ_k+\varepsilon}), \end{equation*} where $C_1(k),\,C_2(k)$ are two constants depending only on $k,$ with $δ_3=\frac{19}{60},\,δ_4=\frac{5}{24},\,δ_5=\frac{19}{140},\,δ_6=\frac{25}{192},\, δ_7=\frac{457}{4032},\,δ_k=\frac{1}{k+2}+\frac{1}{2k^2(k-1)}$ for $k\geqslant8.$

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