Pairs of integers which can be written as the sum of two squares (Q1820813)

Let M be the set of all positive integers that are expressible as the sum of two squares of integers and \(b(.)\) the characteristic function of M. \textit{C. Hooley} [J. Reine Angew. Math. 267, 207-218 (1974; Zbl 0283.10031)] proved \[ \sum_{n\leq x}b(n)b(n+k)>A(k)\frac{x}{\log x} \] where \[ A(k)=\prod_{p| k,p\equiv 3 mod 4}(1+1/p). \] For simplicity the proof is only given for \(k=1\) in detail. In this paper \[ \sum_{x<n\leq x+x^{\theta}}b(n)b(n+k)\asymp A(k)\frac{x^{\theta}}{\log x} \] is proved for \(5/6<\theta \leq 1\) and \(1\leq k\leq x\). The proof uses Hooley's method with other ''weights'' \(\rho\) (n)\(\leq b(n)\) for the lower bound. It should be mentioned that \textit{G. Bantle} [Math. Z. 189, 561-570 (1985; Zbl 0545.10029); Acta Arith. 46, 313-329 (1986; Zbl 0529.10039)] proved upper and lower bounds for sums of the type \[ \sum_{x\leq n<x+x^{\theta},n\equiv 1 mod q}b(n)b(n+1). \]