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

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}^{\inft...

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Bibliographic Details
Published in:arXiv.org 2012-04
Main Author: Nayebi, Aran
Format: Article
Language:English
Subjects:
Integers
Upper bounds
Online Access:Get full text
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Summary: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 \(0< \delta_1 \le 0.000025\).
ISSN:2331-8422