### What is the Limit of a Sequence of Rational Numbers?

Winstein Vilas#### Budapest Graduate Seminar

on 3/9/22

In these lectures we will discuss the venerable problem of representing an integer n as a sum of k ≥ 2 squares of integers (k = 2, 3 or 4). In the first part of the talk we will provide a fairly complete proofs of the 2-squares Theorem (Fermat), the 4-squares theorem (Lagrange) and of the 3-squares theorem (Gauss/Legendre). In the second part, we will describe the shape of the set of such representations (when viewed as a set of vectors on the sphere of radius √ n in R^k ). This will lead us to the theory of modular forms, their associated L-functions and if times permits to interesting dynamical systems.