Ergodic properties of flows for integral points on some hyperboloids in connection with the hypothesis for the Dirichlet $L$–function

This work is dedicated to the study of connection of distribution theory of integral points on the simplest hyperboloid with some hypotheses for Dirichlet $L$-function. In application of discrete ergodic method (further DEM), developed by U. V. Linnik (see [1, 2]) to the problem of distribution of integral points on hyperboloids $x_1 x_3 - x_2^2 = m$ (as well as and in case of sphere) in formulations of theorems about asymptotically even distribution of integral points some auxiliary prime number $p$ such as that symbol of Legendre $\left(\frac{-m}{p}\right)=1$. In ergodic theorems and theorems of mixing for integral points the presence of such simple number was natural as it resulted a flow of primitive points used in DEM in conclusion of asymptotic formulae for numbers of integral points on the sphere and on hyperboloid. The receipt (receiving) of residual members in asymptotic formulae for integral points on areas on the sphere and on hyperboloid in frames of usage DEM (see [2, 3]) is of great interest. Studies in this direction for integral points on ellipsoids were carried out by A. V. Malyshev and by author [3] as well as by E. P. Golubeva [4, 5] by means of method of A. U. Vinogradov [6] which are elaboration of dispersions method of U. V. Linnik [7]. It appears that some weakened hypotheses for Dirichlet $L$–function, directly following from broadened hypotheses of Riman allows to eliminate the mentioned lack. Taking into account that circumstance in combination with that done by A. V. Malyshev and B. M. Shirikov in [8].
There obtained a new proof of key lemma DEM for hyperboloids of both kinds, we give corresponding investigation. In Our work the investigation is done at once for both cases of the simplest hyperboloids and in combination with the use of some hypothesis about the behaviour of Dirichlet $L$–function and obtain considerable simplification of arguments in results.
In connection with our investigation we also note that by the Duke method of modular forms with application of Ivants results [10] we shall obtain asymptotic formulae with absolute residual member for numbers of integral points in areas on the simplest hyperboloid. But in [9] as distinct from our work the distribution of integral points according to classes of deductions according to the given module was not considered. In this connection there appears an interesting problem about transference of Duke's results [9] to the distribution of integral points of the simplest hyperboloid according to to progressions, i.e. according to classes of deductions.
Bibliography: 17 titles.