On a problem of Malyshev A. V. of integer points on multidimensional hyperboloids

R. A. Dokhov

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 209-218.

Résumé

Some development early conducted investigations on the problem of Malyshev A. V. about the number of integer points lying in some areas on multidimensional hyperboloids is given in this work. The task of obtaining of asymptotic formulae for quantity of integer points in areas of the kind of De Luri on multidimensional hyperboloids is put by Malyshev A. V. [1]. De Luri [3] in case of four-dimensional hyperbolic surface \begin{equation*} p\left(x_1, \ldots, x_4\right) = \sum\limits_{k=1}^{4} a_k x_k^2 -m =0, \quad m \ne 0 \end{equation*} in the area $\Omega_p (L)$ on it by defined inequality \begin{equation*} \sum\limits_{k=1}^{4} \left| a_k \right| x_k^2 \leqslant L \end{equation*} obtained asymptotic formula (in $L\to\infty$ and fixed $a_1, a_2, a_3, a_4$, and $m$) for value of $R \left( \Omega_p (L) \right)$, equaled to the number of integer points in the area $\Omega_p (L)$ on the mentioned hyperboloid, but in so doing De Luri does not value the remainder formula. Later on in [1] generalization of this value is given on multidimensional hyperboloid given by the equation \begin{equation*} p = p\left(x_1, \ldots, x_s\right) = \sum\limits_{k=1}^{s} a_k x_k^2 + \sum\limits_{k=1}^{s} b_k x_k + c = 0, \end{equation*} where $a_k$, $b_k$, $(k = 1, \ldots, s)$, $c \ne 0$ — integers, in addition to coefficients $a_k$ not all is one sign, but area of $\Omega_p (L)$ on this hyperboloid is given by the inequality \begin{equation*} \sum\limits_{k=1}^{s} \left| a_k \right| x_k^2 \leqslant L. \end{equation*} In development of indicated task of Malyshev A. V. we examine arbitrary quadratic form equivalent to the diagonal in the equation of hyperboloid, and the area of \begin{equation*} \Omega_p (L) : \sum\limits_{k=1}^{s} \left| a_k \right| x_k^2 \leqslant L \end{equation*} is substituted for the area \begin{equation*} \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\} \leqslant L, \end{equation*} where $Q_i^{(1)}$ и $Q_i^{(2)}$ — binary quadratic forms, equivalent to diagonal forms. In conclusion of our asymptotic result about quantity of $R \left( \Omega_p, L \right)$ the theorem about weighted number of integer points $I_h (n, s)$ from [2] is used in $n\to\infty$ and the complex variant of tauberian’s theorem with remainder term for the power series (see [5, 6]). Also wee will note that our obtained result is analogous to one result of Davenport [7] by generalized problem of Varing in power $k=2$, but in such meaning of $k$ our question of hyperbolic surface has several more common kind. Bibliography: 16 titles.

Détail

Keywords: A. V. Malyshev problem, integer number, multidimensional hyperboloid, quadratic forms, tauberian theorem, asymptotic formula.

Bibliographie

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