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On the sphere problem. (English) Zbl 0837.11054

Let \(S(R)\) be the number of lattice points in the three-dimensional sphere of radius \(R\) and center at the origin. Then an approximate formula \[ S(R) = \frac{4\pi}{3} R^3 + O(R^\vartheta) \] holds with some exponent \(\vartheta\). The so-called sphere problem consists in finding \(\vartheta\) as small as possible. Let \(\vartheta_3\) be the least number such that the approximation is true with any \(\vartheta > \vartheta_3\). Then it is known that \(1 \leq \vartheta_3 \leq \frac{4}{3}\). The upper bound was proved by I. M. Vinogradov [Izv. Akad. Nauk SSSR, Ser. Mat. 27, 957–968 (1963; Zbl 0116.03901)] and Chen Jingrun [Sci. Sin. 12, 751–764 (1963; Zbl 0127.27503)]. The authors now prove \(\vartheta_3 \leq 29/22\).
The method of proof is based on the idea of considering the problem on the one hand as a lattice point problem and on the other hand as a problem of estimating character sums.

MSC:

11P21 Lattice points in specified regions
11N37 Asymptotic results on arithmetic functions
11L40 Estimates on character sums
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