About the modern problems of the theory of hyperbolic zeta-functions of~lattices

N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 176-190.

Résumé

The article gives an expanded version of the report, the author of Made in January 30, 2015 in Moscow at an international conference, dedicated to the memory of Professor A. A. Karatsuba, held at the Mathematical Institute. Russian Academy of Sciences and Moscow State University named after M. V. Lomonosov. The report sets out the facts from the history of the theory of hyperbolic zeta function, provides definitions and notation. The main content of the report was focused discussion of actual problems of the theory of hyperbolic zeta function of lattices. Identified the following promising areas of current research: The problem of the correct order of decreasing hyperbolic zeta function in $ \alpha \to \infty $; The problem of existence of analytic continuation in the left half-plane $ \alpha = \sigma + it \, (\sigma \le1) $ hyperbolic zeta function of lattices $ \zeta_H (\Lambda | \alpha) $; Analytic continuation in the case of lattices S. M. Voronin $ \Lambda (F, q) $; Analytic continuation in the case of joint lattice approximations; Analytic continuation in the case of algebraic lattices   $ \Lambda (t, F) = t \Lambda (F) $; Analytic continuation in the case of an arbitrary lattice $ \Lambda$; The problem behavior hyperbolic zeta function of lattices   $ \zeta_H (\Lambda | \alpha) $ in the critical strip; The problem of values of trigonometric sums grids. As a promising method for investigating these problems has been allocated an approach based on the study of the possibility of passing to the limit by a convergent sequence of Cartesian grids. Bibliography: 19 titles.

Détail

Keywords: lattice, hyperbolic zeta function of lattice, net, hyperbolic zeta function of net, quadrature formula, parallelepiped net, method of optimal coefficients.

Bibliographie

[1] Bakhvalov N. S., “On an approximate calculation of multiple integrals”, Vestnik Moskovskogo universiteta, 1959, no. 4, 3–18 (Russian)

[2] Voronin S. M., “On quadrature formulas”, Russian Acad. Sci. Izv. Math., 45:2 (1995), 417–422

[3] Voronin S. M., “On the construction of quadrature formulas”, Izv. Math. Vol., 59:4 (1995), 665–670 | DOI

[4] Voronin S. M., Temirgaliev N., “Quadrature formulas that are connected with divisors of the field of Gaussian numbers”, Math. Notes, 46:2 (1990), 597–602 | DOI

[5] Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol'skii N. M., Dobrovolsky N. N., Multidimensional Number-theoretic grid and lattice algorithms for finding the optimal coefficients, Izd-vo Tul. state PED. University n.a. L. N. Tolstoy, Tula, 2012, 283 pp. (Russian)

[6] Dobrovolskaya L. P., Dobrovol'skii N. M., Dobrovolsky N. N., Ogorodnichuk N. K., Rebrov E. D., Rebrova I. Yu., “Some questions Number-theoretic methods in approximate analysis”, Proceedings of the X international conference, Scientific notes, Orel state University. Ser. Natural, technical and medical science, no. 6(2), 2012, 90–98 (Russian)

[7] Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol'skii N. M., Dobrovolsky N. N., “Hyperbolic Zeta-function grids and sheets the calculation of the optimal coefficients”, Chebyshevskii Sb., 13:4(44) (2012), 4–107 (Russian)

[8] Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol'skii N. M., Dobrovolsky N. N., “On Hyperbolic Zeta Function of Lattices”, Continuous and Distributed Systems, Solid Mechanics and Its Applications, 211, 2014, 23–62 | DOI

[9] Dobrovol'skii M. N., “A functional equation for the hyperbolic zeta function of integer lattices”, Moscow Univ. Math. Bull., 62:5 (2007), 186–191 | DOI

[10] Dobrovol'skii N. M., Hyperbolic Zeta function lattices, Dep. v VINITI 24.08.84, No 6090-84, 1984 (Russian)

[11] Dobrovol'skii N. M., Van'kova V. S., Kozlova S. L., Hyperbolic Zeta-function of an algebraic lattices, Dep. v VINITI 12.04.90, No 2327-B90, 1990 (Russian)

[12] Dobrovol'skii N. M., Dobrovol'skii N. N., “About some problems of number-theoretic methods in approximate analysis”, Algebra and number theory: modern problems and applications, Proceedings of the XII international. Conf., dedicated to the 80th anniversary V. N. Latysheva, Izd-vo Tul. state PED. University n.a. L. N. Tolstoy, Tula, 2014, 23–27 (Russian)

[13] Dobrovol'skii N. N., “Discrepancy of two-dimensional Smolyak grids”, Chebyshevskii Sb., 8:1(21) (2007), 110–152 (Russian)

[14] Korobov N. M., “Approximate evaluation of repeated integrals”, Dokl. Akad. Nauk SSSR, 124 (1959), 1207–1210 (Russian)

[15] Korobov N. M., “Computation of multiple integrals by the method of optimal coefficients”, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him., 1959, no. 4, 19–25 (Russian)

[16] Temirgaliev N., “Application of the theory of divisors to the numerical integration of periodic functions of several variables”, Math. USSR-Sb., 69:2 (1991), 527–542 | DOI

[17] Fel'dman N. I., Approximations of algebraic numbers, Moskov. Gos. Univ., M., 1981, 200 pp. (Russian)

[18] Frolov K. K., “Upper bounds for the errors of quadrature formulae on classes of functions”, Dokl. Akad. Nauk SSSR, 231:4 (1976), 818–821 (Russian)

[19] Frolov K. K., Kvadraturnye formuly na klassakh funktsiy, PhD thesis, VTS AN SSSR, M., 1979 (Russian)