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

TL;DR

This paper establishes upper bounds on the number of representations of integers as the sum of a prime and a square, advancing understanding of Hardy-Littlewood conjecture limitations.

Contribution

It provides the first known upper bounds for the number of such representations for all integers up to N, including bounds dependent on the existence of a Siegel zero.

Findings

First upper bound applies to all n ≤ N.

Second upper bound depends on the existence of a Siegel zero.

At most a small proportion of integers in the range have more than the bound.

Abstract

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 $0< \delta_1 \le 0.000025$.