Arithmetic progressions of squares and multiple Dirichlet series (Q6581847)

In this paper under review, the authors study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. This multiple Dirichlet series is defined by \N\[\N\mathcal{D}(s, w)=\sum_{\substack{m, h >1\\ (m, h)=1}}\frac{r_1(h)r_1(m)r_1(2m-h)}{m^sh^w}, \N\]\Nwhere \(r_\ell (n)\) denotes the number of ways to represent \(n\) as a sum of \(\ell\) squares. The principal result of this paper is Theorem 5.1, which states that \(\mathcal{D}(s, w)\) has meromorphic continuation to \(\mathbb{C}^2\) by means of spectral expansion. The authors then exploit this meromorphic continuation to obtain a variety of asymptotic results for the distribution of primitive arithmetic progressions of squares and rational points on \(x^2+y^2=2\).