On the number of lattice points of linear comparison solutions in rectangular areas
https://doi.org/10.22405/2226-8383-2022-24-4-311-324
Abstract
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.
In 2002, V. A. Bykovsky obtained fundamentally new estimates from below and from above, which coincided in order.
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—Korobov—Bykovsky theorem on the estimate of the hyperbolic zeta function of the lattice of linear comparison solutions.
The difference between the theorem on the number of lattice points of solutions to linear comparison in rectangular areas and the corresponding Korobov lemma is that instead of an estimate through the ratio of the volume of a rectangular area to the hyperbolic parameter, a modified Bykovsky estimate is given through minimal solutions to linear comparison.
The use of the theorem on the number of lattice points of solutions to linear comparison in rectangular domains is supplemented by the generalized Korobov lemma on estimates of the residual series and a number of other modifications in the proof of the Bakhvalov—Korobov theorem, which made it possible to prove the strengthened Bakhvalov—Korobov—Bykovsky theorem on estimates hyperbolic zeta function of the lattice of linear comparison solutions.
Keywords
parallelepipedal grid,
quadrature formulas,
method of optimal coefficients,
quantitative measure of grid quality
Supporting agencies: The study was carried out within the framework of state task No. 073-03-2022-117/7 on the topic “Numbertheoretic methods in approximate analysis and their applications in mechanics and physics” .
About the Authors
Nadezhda Konstantinovna Ter-Gukasova
National Research University Higher School of Economics
Russian Federation
Russian Federation
HR Clerk of the HR Administration Department
of the Higher School of Economics Personnel Department
Mikhail Nikolaevich Dobrovolsky
Geophysical Center of the Russian Academy of Sciences
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, senior
researcher
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Mikhailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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Review
For citations:
Ter-Gukasova N.K., Dobrovolsky M.N., Dobrovol’skii N.N., Dobrovol’skii N.M. On the number of lattice points of linear comparison solutions in rectangular areas. Chebyshevskii Sbornik. 2023;24(4):311-324. (In Russ.) https://doi.org/10.22405/2226-8383-2022-24-4-311-324
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