The Sum of Four Squares Over Real Quadratic Number Fields

Abstract: Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is $\mathbb Q(\sqrt{5})$. We provide explicit formulas for $r(m)$ for $K= \mathbb Q (\sqrt{2})$ and $K= \mathbb Q(\sqrt{5})$. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bounds for the Eisenstein coefficients. Last, we include examples of the complete theta series decomposition of the sum of four squares over $\mathbb Q (\sqrt{3})$, $\mathbb Q (\sqrt{13})$ and $\mathbb Q (\sqrt{17})$.