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 $x\ge 4$ 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.

14 pages

References in corpus (1)

• Sum of one prime and two squares of primes in short intervals