Chebyshevskii Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Chebyshevskii Sbornik, 2022, Volume 23, Issue 1, Pages 83–105
DOI: https://doi.org/10.22405/2226-8383-2022-23-1-83-105
(Mi cheb1156)
 

Generalized Dirichlet problem for a two-dimensional lattice of Dirichlet approximations

N. N. Dobrovol'skiiab, M. N. Dobrovol'skiic, V. N. Chubarikovd, I. Yu. Rebrovaa, N. M. Dobrovol'skiia

a Tula State Lev Tolstoy Pedagogical University (Tula)
b Tula State University (Tula)
c Geophysical centre of RAS (Moscow)
d Lomonosov Moscow State University (Moscow)
References:
Abstract: The paper studies the relationship between the problem of determining the number of points of a two-dimensional lattice of Dirichlet approximations in a hyperbolic cross and the integral representation of the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations. The concept of components of hyperbolic zeta-functions of a two-dimensional lattice of Dirichlet approximations is introduced. A representation is found for the first component of the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations via the Riemann zeta function. With respect to the first component, the paradoxical fact is established that it is continuous for any irrational $\beta$ and discontinuous at all rational points of $\beta$. This refers to the dependency only on the $\beta$ parameter.
For the second component of the hyperbolic zeta-function of the two-dimensional lattice of Dirichlet approximations in the case of a rational value $\beta=\frac{a}{b}$, an asymptotic formula is obtained for the number of points of the second component of the two-dimensional lattice of Dirichlet approximations in the hyperbolic cross. The resulting formula gives an integral representation in the half-plane $\sigma>\frac{1}{2}$.
The main research tool was the Euler summation formula. For the purposes of the work, it was necessary to obtain explicit expressions of the residual terms in asymptotic formulas for the number of points of residue classes of a two-dimensional lattice of Dirichlet approximations over a stretched fundamental lattice $b\mathbb{Z} \times \mathbb{Z}$. Both Theorem 1 and Theorem 2, proved in the paper, show the dependence of the second term of the asymptotic formula and the deduction of the hyperbolic zeta function of the lattice $\Lambda\left(\frac{a}{b}\right)$ depends on the magnitude of the denominator $b$ and independence from the numerator $a$. Earlier, similar effects were discovered by A. L. Roscheney for other generalizations of the Dirichlet problem. The paper sets the task of clarifying the order of the residual term in asymptotic formulas by studying the quantities
$$ R_1^*(T,b,\delta)=\sum_{q=1}^{\frac{\sqrt{T}}{b}}\left\{\frac{T}{bq}-\delta\right\}-\frac{\sqrt{T}}{2b}, R_2^*(T,b,\delta)=\sum_{p=1}^{\sqrt{T}-\delta}\left\{\frac{T}{bp+b\delta}\right\}-\frac{\sqrt{T}}{2}. $$

It is proposed to first study the possibilities of the elementary method of I. M. Vinogradov, and then to obtain the most accurate estimates using the method of trigonometric sums. The paper outlines the directions of further research on this topic.
Keywords: riemann zeta function, dirichlet series, hurwitz zeta function.
Funding agency Grant number
Russian Foundation for Basic Research 19-41-710004_р_а
This work was prepared under a grant from the RFBR № 19-41-710004 _r_а. The work was supported financially by a grant from the Government of the Tula Region under Contract ДС/294 dated November 16, 2021.
Received: 24.12.2021
Accepted: 27.02.2022
Document Type: Article
UDC: 511.3
Language: Russian
Citation: N. N. Dobrovol'skii, M. N. Dobrovol'skii, V. N. Chubarikov, I. Yu. Rebrova, N. M. Dobrovol'skii, “Generalized Dirichlet problem for a two-dimensional lattice of Dirichlet approximations”, Chebyshevskii Sb., 23:1 (2022), 83–105
Citation in format AMSBIB
\Bibitem{DobDobChu22}
\by N.~N.~Dobrovol'skii, M.~N.~Dobrovol'skii, V.~N.~Chubarikov, I.~Yu.~Rebrova, N.~M.~Dobrovol'skii
\paper Generalized Dirichlet problem for a two-dimensional lattice of Dirichlet approximations
\jour Chebyshevskii Sb.
\yr 2022
\vol 23
\issue 1
\pages 83--105
\mathnet{http://mi.mathnet.ru/cheb1156}
\crossref{https://doi.org/10.22405/2226-8383-2022-23-1-83-105}
Linking options:
  • https://www.mathnet.ru/eng/cheb1156
  • https://www.mathnet.ru/eng/cheb/v23/i1/p83
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024