Sum of a prime and a square

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

“…Hence, in order to prove an upper bound for R √ m , Σ(n, Q, β), and u β−1 . It follows from Lemma 2.7 that for all N/2 < n ≤ N , with t = r, except for ≪ N 1−δ1 of these integers, when r > N 3.6δ1 [11],…”

“…In order to prove Theorem 1.2, we introduce some auxiliary functions and lemmas about these functions. We should inform the reader that any lemma that is presented without proof in this paper means that it has already been stated and proven by Polyakov [11]. Put…”

“…Brünner, Perelli, and Pintz [9] used the methods of Montgomery and Vaughan [7] to prove the same result. Polyakov [11] independently demonstrated that the results of Vinogradov [2] hold without using Siegel's theorem, thereby enabling him to attain effective results. Basing his methods on those of [9], Wang [13] made the computation of c more rigorous and proved,…”

“…In this paper, the first upper bound we prove for R(n) holds for all n ≤ N (N sufficiently large) with no exceptions. The second upper bound is achieved by way of Polyakov's [11] methods, although he uses his methods for the entirely different purpose of determining the cardinality of the exceptional set E(N ). This second upper bound assumes the possible existence of the Siegel zero, and consequently has an exceptional set.…”

“…Notation. We will use some of the same notation used by Polyakov [11] for simplicity: Suppose m and u are natural numbers; p is a prime; N is a sufficiently large positive integer; µ is the Möbius function; ϕ is Euler's totient function;…”