Hardy 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 \mathcal{ R}(n) \sim \frac{ \sqrt{ n}}{ \log n}\prod_{ p=3}^{ \infty}(1-\frac{ 1}{ p-1}(\frac{ n}{ p})), where $p$ is a prime, $m$ is an integer, and $(\frac{ n}{ p})$ 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 $0< \delta_1 \le 0.000025$.
Categories: math.NT