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Application of spherical functions to a certain problem in the theory of quadratic forms
Authors:E. P. Golubeva  O. M. Fomenko
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 ldquocap.rdquo The precise result is formulated in the following manner: let rell(n) be the number of the representations of n by a sum of ell squares, 0; then for even ellge6 we have for ell=4 we have where n=2agrn1, 2agr par n; the expression for Kell(A), ell ge 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.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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