The sum of four squares over real quadratic number fields

Abstract

Well-known results of Lagrange and Jacobi prove that every [Formula: see text] can be expressed as a sum of four integer squares, and the number [Formula: see text] of such representations can be given by an explicit formula in [Formula: see text]. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is [Formula: see text]. We provide explicit formulas for [Formula: see text] for [Formula: see text] and [Formula: see text]. 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 [Formula: see text], [Formula: see text] and [Formula: see text].