On a theorem of Hua
Let \(E_1(x)\) denote the number of integers \(n\leq x\), \(n\equiv 3(24)\), \(n\not\equiv 0(5)\) which are not representable as \(n= p^2_1+ p^2_2+ p^2_3\), and similarly, \(E_2(x)= \#\{n\leq x\), \(2\mid n\), \(n\not\equiv 1(5)\), \(n\neq p_1+ p^2_2\}\). The bounds \(E_j(x)\ll x(\log x)^{-A}\) (\(j= 1,2\); \(A>0\), arbitrary) were proved by \textit{L.-K. Hua} [Additive Primzahltheorie (1959; Zbl 0088.25503)] and \textit{W. Schwarz} [J. Reine Angew. Math. 206, 78-111 (1961; Zbl 0102.28201)]. In the present article, the authors provide short interval versions: \[ E_j(x+ H)- E_j(x)\ll H(\log x)^{-A},\;x^{\vartheta_j+ \varepsilon}\leq H\leq x, \] where \(\vartheta_1= 3/4\), \(\vartheta_2= 7/16\). The ideas of proof are similar to those used by \textit{A. Perelli} and \textit{J. Pintz} [J. Numer. Theory 54, 297-308 (1995; Zbl 0851.11056)] and \textit{H. Mikawa} [Tsukuba J. Math. 17, 299-310 (1993; Zbl 0802.11037)].
scientific article; zbMATH DE number 1145768