An \(\Omega_+\)-estimate for the number of lattice points in a sphere
Let \(A(t)\) denote the number of lattice points (of the standard lattice \(\mathbb Z^3)\) in the space \(x^2 + y^2 + z^2 \le T\) and put \(P(T) = A(T) - (4\pi/3) T^{3/2}\). It is known after \textit{I. M. Vinogradov} [Izv. Akad. Nauk SSSR, Ser. Mat. 27, 957--968 (1963; Zbl 0116.03901)] that \(P(T) = O(T^{2/3} (\log T)^6)\) and after \textit{G. Szegö} [Math. Z. 25, 388--404 (1926; JFM 52.0175.03)] that \(P(T) = \Omega_{-}(T^{1/2}(\log T)^{1/2})\). In the direction of an \(\Omega_{+-}\) estimate, \textit{K. Chandrasekharan} and \textit{R. Narasimhan} have proved [Acta Arith. 6, 487--503 (1961; Zbl 0101.03703)] that \(\displaystyle \limsup_{T\to\infty} P(T) T^{-1/2} = +\infty\). In this paper the estimate \[ P(T) = \Omega_{+} \left(T^{1/2}(\log_2 T)^{1/2} (\log_3 T)^{-1/2}\right) \] is established \((\log_k\) denotes the \(k\)-fold iterated logarithm). The proof makes use of a method due to \textit{K. S. Gangadharan} [Proc. Camb. Philos. Soc. 57, 699--721 (1961; Zbl 0100.03901)] and of the explicit formula for \(r_3(n)\) (the number of representations of \(n\) as a sum of three integer squares).
scientific article; zbMATH DE number 3821867
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