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

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 R(n) of such representations for n = p + m 2 is asymptotically given bywhere p is a prime, m is an integer, and n p denotes the Legendre symbol. Unfortunately, as we will later point out, this conjecture is difficult to prove and not 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 R(n) for n ≤ N . The first upper bound applies to all n ≤ 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 ≤ N but at most ≪ N 1−δ 1 of these integers, where N is a sufficiently large positive integer and 0 < δ 1 ≤ 0.000025.