Arithmetic progressions of squares and multiple Dirichlet series

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to \(\mathbb {C}^2\) ${\mathbb{C}}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on \(x^2+y^2=2\) $x^2+y^2=2$ .