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- W1888894765 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 $0< delta_1 le 0.000025$." @default.
- W1888894765 created "2016-06-24" @default.
- W1888894765 creator A5058188874 @default.
- W1888894765 date "2010-04-04" @default.
- W1888894765 modified "2023-09-27" @default.
- W1888894765 title "Upper bounds on the solutions to $n = p+m^2$" @default.
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