The number of representations of $n$ as a growing number of squares

Abstract

Let $r_{k}(n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k}(n)$ when $n$ grows linearly with $k$. As a special case, we find that \[ r_{n}(n) \sim \frac{B \cdot A^{n}}{\sqrt{n}}, \] with $B \approx 0.2821$ and $A \approx 4.133$.