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

N. N. Dobrovol'skii ; M. N. Dobrovol'skii ; V. N. Chubarikov ; I. Yu. Rebrova ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 83-105

Résumé

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.

Détail

Keywords: riemann zeta function, dirichlet series, hurwitz zeta function.