## Undergraduate Junior Paper

**Theta Functions: The Problem of Representation of Numbers as Sums of Squares**

Undergraduate Junior Paper, São Paulo State University (UNESP), 2005

Advisor: S. Christodoulou

**Abstract:**In this paper we use techniques developed by Jacobi, Ramanujan (such as the Ramanujan'ssums) and modular functions to prove some theorems on the representation of numbers as sums of squares. We start with a theorem of Jacobi, which allows to obtain formulas for $r_{2}(n)$ and $r_{4}(n)$ (formula for the representation of a number as a sum of two squares and four squares respectively) as corollaries of an analytic theory. We then use some theorems of Ramanujan to deduce an expression for $\vartheta^{4}(x)$. We then studied formulas to determine $r_{2s}(n)$ when $2s$ exceeds $8$ and so we need to use some sums developed by Ramanujan. The use of modular functions and the modular group is required to prove that $r_{8}(s) = \rho_{8}(s)$ and a large part of this paper is concerned to proof this fact. The rest of the paper is concerned to study the issue of representation of numbers as sums of $24$ squares and ends

with a brief discussion of the Ramanujan's function $\tau(n)$ and its relation to the problem of representation of numbers as sums of squares.