An Omega theorem on Pythagorean triples (Q1313454)

Let \(A(x)\) denote the number of Pythagorean triples \((r,s,n)\) with \(r^ 2+ s^ 2= n^ 2\) and \(1\leq n\leq x\). Then \[ A(x)+ {\textstyle {4\over\pi}} x\log x+Bx+ E(x), \] where \(B\) is a well-defined constant. The remainder \(E(x)\) can be estimated by \(E(x)= O(\sqrt{x})\). Somewhat better results can be found in the papers of \textit{M. I. Stronina} [Izv. Vyssh. Uchebn. Zaved Mat. 1969, No. 8(87), 112-116 (1969; Zbl 0222.10054)] and \textit{W. G. Nowak} and \textit{W. Recknagel} [Math. J. Okayama Univ. 31, 213-220 (1989; Zbl 0702.11064)]. Moreover, M. I. Stronina proved \(E(x)= \Omega(x^{1/4})\). In this paper the author gives the substantial improvement \(E(x)= \Omega(x^{1/3})\).