Upper bounds on the solutions to n = p+m^2

Abstract

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by <br />begin{equation*} <br />mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), <br />end{equation*} <br />where $p$ is a prime, $m$ is an integer, and $left(frac{n}{p}right)$ denotes the Legendre symbol. Unfortunately, as we will later point out, this conjecture is difficult to prove and not emph{all} integers that are nonsquares can be represented as the sum of a prime and a square. Instead in this paper we prove two upper bounds for $mathcal{R}(n)$ for $n le N$. The first upper bound applies to emph{all} $n le N$. The second upper bound depends on the possible existence of the Siegel zero, and assumes its existence, and applies to all $N/2 < n le N$ but at most $ll N^{1-delta_1}$ of these integers, where $N$ is a sufficiently large positive integer and $0