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Distribution of Lattice Points over the Four-Dimensional Sphere

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Abstract

Let rl(n) be the number of representations of n by a sum of l squares of integers and let 0 < A < 1 be a constant. It is proved that if (n,2)=1, then \(\Sigma _{ - A \leqslant w/\sqrt n \leqslant A{\text{ }}} r_3 (n - w^2 ) = \mu _4 (A)r_4 (n) + O(n^{1487/2000} ),\mu _4 (A) >0\). Previously, the author obtained this asymptotics with a weaker error term O(\((n^{{3 \mathord{\left/ {\vphantom {3 {4 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {4 + \varepsilon }}} )\). Bibliography: 12 titles.

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REFERENCES

  1. O. M. Fomenko, “On the uniform distribution of integral points on multidimensional ellipsoids,” Zap. Nauchn. Semin. LOMI, 154, 144-153 (1986).

    Google Scholar 

  2. O. M. Fomenko, “Applications of the Petersson formula for a bilinear form of Fourier coefficients of cusp forms,” Zap. Nauchn. Semin. POMI, 204, 143-166 (1993).

    Google Scholar 

  3. I. M. Vinogradov, “On the number of integer points in a ball,” Izv. Akad. Nauk SSSR., Ser. Mat., 27, 957-968 (1963).

    Google Scholar 

  4. Chen Jing-Run, “Improvement on the asymptotic formulas for the number of lattice points in a region of three dimensions. II,” Sci. Sinica, 12, 751-764 (1963).

    Google Scholar 

  5. F. Chamizo and H. Iwaniec, “On the sphere problem,” Revista Mathem. Iberoamericana, 11, 417-429 (1995).

    Google Scholar 

  6. D. R. Heath-Brown, “Lattice points in the sphere,” in: Number Theory in Progress. Proc. Intern. Conf. on Number Theory, Zakopane, Poland, June 30-July 9 (1997). Vol. II, Berlin (1999), pp. 883-891.

    Google Scholar 

  7. E. P. Golubeva, “On Waring's problem for a ternary quadratic form and an arbitrary even power,” Zap. Nauchn. Semin. LOMI, 144, 27-37 (1985).

    Google Scholar 

  8. E. P. Golubeva and O. M. Fomenko, “Application of spherical functions to a certain problem in the theory of quadratic forms,” Zap. Nauchn. Semin. LOMI, 144, 38-45 (1985).

    Google Scholar 

  9. P. T. Bateman, “On the representations of a number as the sum of three squares,” Trans. Am. Math. Soc., 71, 70-101 (1951).

    Google Scholar 

  10. H. L. Montgomery, “Topics in multiplicative number theory,” in: Lect. Notes Math., 227, Springer-Verlag (1971).

  11. V. A. Bykovskii, “Density theorems and the mean value of arithmetical functions on short intervals,” Zap. Nauchn. Semin. POMI, 212, 56-70 (1994).

    Google Scholar 

  12. T. V. Vepkhvadze, “On the analytic theory of quadratic forms,” Tr. Tbiliss. Mat. Inst. Razmadze, 72, 12-31 (1983).

    Google Scholar 

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  1. St.Petersburg Department of the Steklov Mathematical Institute, Russia

    O. M. Fomenko

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Fomenko, O.M. Distribution of Lattice Points over the Four-Dimensional Sphere. Journal of Mathematical Sciences 110, 3164–3170 (2002). https://doi.org/10.1023/A:1015484630940

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