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

TL;DR

This paper derives an asymptotic formula for the number of representations of an integer as a sum of k squares when both n and k grow linearly, revealing exponential growth with specific constants.

Contribution

It provides the first asymptotic estimate for r_k(n) in the regime where n grows linearly with k, including a precise formula for r_n(n).

Findings

Asymptotic formula for r_k(n) when n grows linearly with k

Explicit constants B and A for the case r_n(n)

Demonstrates exponential growth of r_n(n) with n

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 $A \approx 4.133$.