Linear correlations of the divisor function

Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet $L$-functions, we provide analytic continuation for the series $\mathcal A_{\boldsymbol{a}}(s):=\sum_{n_1,\dots,n_k\geq1}\frac{d(n_1)\cdots d(n_k)}{(n_1\cdots n_k)^{s}}$ with the sum restricted to solutions of a non-trivial linear equation $a_1{n}_1+\cdots +a_k{n}_k=0$. The series $\mathcal A_{\boldsymbol{a}}(s)$convergesabsolutelyfor$\Re(s)>1-\frac1k$ and we show it can be meromorphically continued to $\mathrm{\Re }(s)>1-\frac{2}{k+1}$ with poles at $s=1-\frac{1}{k-j}$ only, for $1\le j<(k-1)\mathrm{/}2$. As an application, we obtain an asymptotic formula with power saving error term for the number of points in the variety $a_1{x}_1{y}_1+\cdots +a_k{x}_k{y}_k=0$ in ${\mathbb{P}}^{k-1}(\mathbb{Q})\times {\mathbb{P}}^{k-1}(\mathbb{Q})$.