# Sums of squares in real quadratic fields and Hilbert modular groups

## TL;DR

This paper employs spectral theory of Hilbert-Maass forms for real quadratic fields to analyze the asymptotic behavior of sums related to representations as sums of two squares in the ring of integers.

## Contribution

It introduces a novel application of spectral theory to study sums of two squares in real quadratic fields, providing new asymptotic results.

## Key findings

- Derived asymptotic formulas for sums of representations as sums of two squares.
- Connected spectral theory of Hilbert-Maass forms with classical number theory problems.
- Extended understanding of sums of squares in the context of real quadratic fields.

## Abstract

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.