On the weighted number of integer points on some multidimensional hyperboloids

R. A. Dokhov ; U. M. Pachev

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 219-245.

Résumé

In this paper asymptotic formula for weighted number of integer points on multidimensional hyperbolic surfaces defined by direct sum of indefinite quaternary integral quadratic forms of singular kind is obtained. In doing so weighted function is chosen as a real exponent on the index of which there stands integral quadratic form being direct sum of positive binary quadratic forms with the same discriminant equal to the discriminant $\delta_{F}$ of imaginary quadratic field $F = \Theta \left( \sqrt{d} \right)$ where $d$ is the negative without quadrate number. The choice of real kind of weighting function is conditioned by possibility application used method in investigation of question about the number of integer points lying is some fields of real kind on examining multidimensional hyperboloids. Leaning upon the method of article [7] based on the use of exact meanings of Gauss double sum we examine multidimensional problem about weighted number of integer points on hyperbolic surface of real kind. The question is about the asymptotic with remainder of series for value $$ I_{h} \left(n, s \right) = \sum\limits_{p\left(\overline{x},\overline{y},\overline{z},\overline{t}\right) = h} { e^{-\frac{\omega\left(\overline{x},\overline{y},\overline{z},\overline{t}\right) }{n}} }, $$ where $n \to \infty$ — real parameter, $$ p\left(\overline{x},\overline{y},\overline{z},\overline{t}\right) = \sum\limits_{i = 1}^{s} \left\{ Q_i^{(1)}\left( {x_i, y_i} \right) - Q_i^{(2)}\left( {z_i, t_i} \right) \right\}, $$ $$ \omega\left(\overline{x},\overline{y},\overline{z},\overline{t}\right) = \sum\limits_{i = 1}^{s} \left\{ Q_i^{(1)}\left( {x_i, y_i} \right) + Q_i^{(2)}\left( {z_i, t_i} \right) \right\}, $$ $Q_i^{(1)}, Q_i^{(2)}$ — positive integral binary quadratic forms of the same discriminant $\delta_{F}$; $h \ne 0$ — integral number. In deducing the asymptotic formula for $I_{h} \left(n, s \right)$ essentially we use: 1) the formula of turning of theta-series binary quadratic form (in our case it is enough to use double theta-series instead of multidimensional); 2) formula for $$ \int\limits_{- \frac{1}{q(q+N)}}^{\frac{1}{q(q+N)}} { \frac{e^{-2\pi i h x}}{\left( \frac{1}{n^2} + 4 \pi^2 x^2 \right)^S} } dx $$ 3) estimation of sum of Kloosterman $$ K \left( {u, v; q} \right) = {\sum\limits_{x\, \text{mod}\, q}}^{\prime} e^{\frac{2 \pi i}{q} \left( ux + vx^{'} \right)}, $$ where $xx^{'} \equiv 1\, \left( \text{mod}\, q \right)$. Obtained asymptotic formula for $I_{h} \left(n, s \right)$ generalises one of the results of Kurtova L. N. [7] about weighted number of integer points on four-dimensional hyperboloids for the case of multidimensional hyperboloids corresponding real kind. Besides our result in case of constant coefficients of hyperboloid equation also generalized one result of Malishev A. B. [10] for a case of some nondiagonal quadratic forms in comparison with the result of Golovizina V. V. [3] the main number in examining problem is obtained in evident kind as in our work exact meanings of Gauss double sums are used and in [3] it is expressed by way of some complex integral $W(N)$, for which only estimation is given over in doing so in our case $N = \left[ \sqrt{n} \right]$. Later on the result about value $I_{h} \left(n, s \right)$ can be applied in obtaining asymptotic formulae for the number of integer points lying in some fields of real kind on multidimensional hyperboloids. Bibliography: 16 titles.

Détail

Keywords: circle method, weighted number of integer points, hyperbolic surface, multidimensional hyperboloid, asymptotic formula, quadratic forms, theta-series of quadratic form, Gauss double sum, Klosterman sum.

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