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Application of spherical functions to a certain problem in the theory of quadratic forms

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Abstract

The problem regarding the number of integral points on multidimensional ellipsoids is investigated with the aid of modular forms. In the paper we consider the simplest special case of the following problem: one considers a multidimensional sphere and as a domain on it one selects a “cap.” The precise result is formulated in the following manner: let rℓ(n) be the number of the representations of n by a sum of ℓ squares, 0<A<ℓ; then for even ℓ≥6 we have

for ℓ=4 we have

where n=2αn1, 2α ∥ n; the expression for Kℓ(A), ℓ ≥ 4, is given in the paper. It is also shown that one can refine somewhat the results on the distribution of integral points on multidimensional ellipsoids, obtained by A. V. Malyshev by the circular method, remaining within the framework of the same methods.

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Literature cited

  1. A. V. Malyshev, “Representation of integers by positive quadratic forms,” Trudy Mat. Inst. Akad. Nauk SSSR,65 (1962).

  2. C. Pommerenko, “über die Gleichverteilung von Giotterpunkten auf m-dimensionalen Ellipsoiden,” Acta Arith.,5, No. 2, 227–257 (1959).

    Google Scholar 

  3. E. V. Podsypanin, “The number of integral points in an elliptic region (a remark on theorem of A. V. Malyshev),” J. Sov. Math.,18, No. 6 (1982).

  4. H. Batman and A. Erdelyi, Higher Transcendental Functions, Vols. I and II, McGraw-Hill, New York (1953).

    Google Scholar 

  5. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ Press (1971).

  6. E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen (1959).

    Google Scholar 

  7. O. M. Fomenko, “Estimates of the Petersson inner product with application to the theory of quaternary quadratic forms,” Dokl. Akad. Nauk SSSR,152, No. 3, 559–562 (1963).

    Google Scholar 

  8. A. N. Podkorytov, “On the Lebesgue constants of double Fourier series,” Vestn. Leningr. Gos. Univ., Mat. Mekh. Astron., No. 7, 79–84 (1977).

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Authors
  1. E. P. Golubeva
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  2. O. M. Fomenko
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 38–45, 1985.

We express our gratitude to M. A. skopina for a consultatin regarding the theory of multiple fourier series

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Golubeva, E.P., Fomenko, O.M. Application of spherical functions to a certain problem in the theory of quadratic forms. J Math Sci 38, 2054–2060 (1987). https://doi.org/10.1007/BF01474438

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  • DOI: https://doi.org/10.1007/BF01474438

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