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

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 $X\ge4$ for which this property does not hold. 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 on the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claim $A=4,3$ respectively.