A remark on the conditional estimate for the sum of a prime and a square

TL;DR

This paper improves the conditional estimate on the number of integers that do not conform to Hardy and Littlewood's conjecture, showing a tighter bound under the Generalized Riemann Hypothesis.

Contribution

It provides a refined upper bound for the count of exceptions to the conjecture, improving previous estimates under GRH.

Findings

Established a bound of $E(x) \,\ll\, x^{1/2}(\,\log x)^A(\,\log\log x)^4$ with $A=3/2$

Improved previous bounds of $A=4$ and $A=3$ by Mikawa and Perelli-Zaccagnini

Confirmed the conjecture holds for almost all large integers under the hypothesis

Abstract

Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(x)$ be the number of positive integers up to $x\ge4$ which does not satisfy this condition. We prove $E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement of the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claims $A=4,3$ respectively.