The Sum of Four Squares Over Real Quadratic Number Fields

TL;DR

This paper investigates the universality of representing numbers as sums of four squares over real quadratic fields, identifying the unique field where this is universal, and providing explicit formulas and bounds for related theta series.

Contribution

It proves that only the field Q(√5) allows universal sums of four squares among real quadratic fields and derives explicit formulas and bounds for the associated theta series.

Findings

Q(√5) is the only real quadratic field with universal four squares sums

Explicit formulas for representation counts r(m) over Q(√2) and Q(√5)

Bounds for Eisenstein coefficients of theta series

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 $\mathbb Q (\sqrt{17})$.