On the number of lattice points of linear comparison solutions in rectangular areas
Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 130-144
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In the theory of the hyperbolic zeta function of lattices, a significant role is played by the Bakhvalov theorem, in which the magnitude of the zeta function of the lattice of linear comparison solutions is estimated through the hyperbolic lattice parameter.
In N. M. Korobov's 1963 monograph, this theorem is proved by a method different from the original work of N. S. Bakhvalov. In this method, the central role is played by the lemma about the number of linear comparison solutions in a rectangular area.
The paper gives new estimates of the number of lattice points of linear comparison solutions in rectangular regions. This allows us to prove the strengthened Bakhvalov theorem on the evaluation of the hyperbolic zeta function of the lattice of solutions of linear comparison.
The difference between the theorem on the number of lattice points of linear comparison solutions in rectangular regions and the corresponding Korobov lemma is that instead of one estimate through the ratio of the volume of a rectangular region to a hyperbolic parameter, two more cases are added and in the first case the constant is reduced. The use of the theorem on the number of lattice points of linear comparison solutions in rectangular areas leads to the need to prove the Bakhvalov–Korobov theorem to consider various areas of application of the theorem on the number of lattice points of linear comparison solutions in rectangular areas.
Keywords:
parallelepipedal grid, quadrature formulas, method of optimal coefficients, quantitative measure of grid quality.
@article{CHEB_2022_23_5_a10,
author = {N. K. Ter-Gukasova and M. N. Dobrovol'skii and N. N. Dobrovol'skii and N. M. Dobrovol'skii},
title = {On the number of lattice points of linear comparison solutions in rectangular areas},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {130--144},
publisher = {mathdoc},
volume = {23},
number = {5},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a10/}
}
TY - JOUR AU - N. K. Ter-Gukasova AU - M. N. Dobrovol'skii AU - N. N. Dobrovol'skii AU - N. M. Dobrovol'skii TI - On the number of lattice points of linear comparison solutions in rectangular areas JO - Čebyševskij sbornik PY - 2022 SP - 130 EP - 144 VL - 23 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a10/ LA - ru ID - CHEB_2022_23_5_a10 ER -
%0 Journal Article %A N. K. Ter-Gukasova %A M. N. Dobrovol'skii %A N. N. Dobrovol'skii %A N. M. Dobrovol'skii %T On the number of lattice points of linear comparison solutions in rectangular areas %J Čebyševskij sbornik %D 2022 %P 130-144 %V 23 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a10/ %G ru %F CHEB_2022_23_5_a10
N. K. Ter-Gukasova; M. N. Dobrovol'skii; N. N. Dobrovol'skii; N. M. Dobrovol'skii. On the number of lattice points of linear comparison solutions in rectangular areas. Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 130-144. http://geodesic.mathdoc.fr/item/CHEB_2022_23_5_a10/