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This article is cited in 2 scientific papers (total in 2 papers)
On the weighted number of integer points on some multidimensional hyperboloids
R. A. Dokhov, U. M. Pachev Kabardino-Balkar State University, Nal'chik
Abstract:
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.
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.
Received: 29.07.2015
Citation:
R. A. Dokhov, U. M. Pachev, “On the weighted number of integer points on some multidimensional hyperboloids”, Chebyshevskii Sb., 16:3 (2015), 219–245
Linking options:
https://www.mathnet.ru/eng/cheb416 https://www.mathnet.ru/eng/cheb/v16/i3/p219
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