Hardy-Littlewood numbers in short intervals
A positive integer \(n\) is called a Hardy-Littlewood number if it can be written as a sum of a prime and a square. A quantitative corollary of the authors' main theorem is that for \(\varepsilon\), \(A > 0\), \(N^{7/24 + \varepsilon} \leq H \leq N\), all but \(O(H (\ln N)^{- A})\) integers \(n \in \lfloor N,N + H\rfloor\) are Hardy-Littlewood numbers. The proof follows the lines of earlier papers on similar questions (the first jointly with \textit{R. Brünner}) [Acta Math. Hung. 53, 347-365 (1989; Zbl 0683.10039); J. Lond. Math. Soc., II. Ser. 47, 41-49 (1993; Zbl 0806.11042)]. The authors state that the same result was proved independently by H. Mikawa.
scientific article; zbMATH DE number 809126