Joint discrete approximation of a pair of analytic functions by periodic zeta-functions
Abstract
In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.
Keyword : Hurwitz zeta-function, non-trivial zeros of the Riemann zeta-function, periodic zeta-function, periodic Hurwitz zeta-function, universality
How to Cite
Balčiūnas, A., Garbaliauskienė, V., Karaliūnaitė, J., Macaitienė, R., Petuškinaitė, J., & Rimkevičienė, A. (2020). Joint discrete approximation of a pair of analytic functions by periodic zeta-functions. Mathematical Modelling and Analysis, 25(1), 71-87. https://doi.org/10.3846/mma.2020.10450
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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V. Franckevič, A. Laurinčikas and D. Šiaučiūnas. On approximation of analytic functions by periodic Hurwitz zeta-functions. Math. Modell. Analysis, 24(1):20– 33, 2019. https://doi.org/10.3846/mma.2019.002
V. Garbaliauskienė, J. Karaliūnaitė and R. Macaitienė. On discrete value distribution of certain compositions. Math. Modell. Analysis, 24(1):34–42, 2019. https://doi.org/10.3846/mma.2019.003
R. Garunkštis and A. Laurinčikas. Discrete mean square of the Riemann zetafunction over imaginary parts of its zeros. Periodica Math. Hung., 76(2):217–228, 2018. https://doi.org/10.1007/s10998-017-0228-6
R. Garunkštis and A. Laurinčikas. The Riemann hypothesis and universality of the Riemann zeta-function. Math. Slovaca, 68(4):741–748, 2018. https://doi.org/10.1515/ms-2017-0141
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181(2):127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
A. Javtokas and A. Laurinčikas. Universality of the periodic Hurwitz zeta-function. Integral Transf. Spec. Functions, 17:711–722, 2006. https://doi.org/10.1080/10652460600856484
J. Kaczorowski. Some remarks on the universality of periodic L-functions. In R. Steuding and J. Steuding(Eds.), New directions in value-distribution theory of zeta and L-functions, pp. 113–120. Shaker Verlag, Aachen, 2009.
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A. Laurinčikas. Joint universality for periodic Hurwitz zeta-functions. Izv. Math., 72:741–760, 2008. https://doi.org/10.1070/IM2008v072n04ABEH002421
A. Laurinčikas. Joint universality of zeta-functions with periodic coefficients. Izv. Math., 74(3):515–539, 2010. https://doi.org/10.1070/IM2010v074n03ABEH002497
A. Laurinčikas. Universality of composite functions of periodic zeta-functions. Sb. Math., 203:1631–1643, 2012. https://doi.org/10.1070/SM2012v203n11ABEH004279
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A. Laurinčikas. Universality theorems for zeta-functions with periodic coefficients. Sb. Math. J., 57:330–339, 2016. https://doi.org/10.1134/S0037446616020154
A. Laurinčikas. A discrete version of the Mishou theorem. II. Proc. Steklov Inst. Math., 296(1):172–182, 2017. https://doi.org/10.1134/S008154381701014X
A. Laurinčikas. On discrete universality of the Hurwitz zeta-functions. Results Math., 72(1-2):907–917, 2017. https://doi.org/10.1007/s00025-017-0702-8
A. Laurinčikas. Joint value distribution theorems for the Riemann and Hurwitz zeta-functions. Moscow Math. J., 18(2):349–366, 2018. https://doi.org/10.17323/1609-4514-2018-18-2-349-366
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. Quaest. Math., 42(5):687–699, 2019. https://doi.org/10.2989/16073606.2018.1481891
A. Laurinčikas. Non-trivial zeros of the Riemann zeta-function and joint universality theorems. J. Math. Anal. Appl., 475(1):395–402, 2019. https://doi.org/10.1016/j.jmaa.2019.02.047
A. Laurinčikas and R. Macaitienė. The discrete universality of the periodic Hurwitz zeta-function. Integral Transf. Spec. Functions, 20:673–686, 2009. https://doi.org/10.1080/10652460902742788
A. Laurinčikas and R. Macaitienė. Joint approximation of analytic functions by shifts of the Riemann and periodic Hurwitz zeta-functions. Appl. Anal. Discrete Math., 12(2):508–527, 2018. https://doi.org/10.2298/AADM170713016L
A. Laurinčikas, R. Macaitienė, D. Mochov and D. Šiaučiūnas. Universality of the periodic Hurwitz zeta-function with rational parameter. Siber. Math. J., 59(5):894–900, 2018. https://doi.org/10.1134/S0037446618050130
A. Laurinčikas and D. Mochov. A discrete universality theorem for the periodic Hurwitz zeta-functions. Chebysh. Sb., 17:148–159, 2016.
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210, 2019. https://doi.org/10.1070/SM9194
A. Laurinčikas and D. Šiaučiūnas. Remarks on the universality of periodic zeta-functions (in Russian). Math. Notes., 80(3-4):532–538, 2006. https://doi.org/10.1007/s11006-006-0171-y
R. Macaitienė, M.Stoncelis and Šiaučiūnas. A weighted discrete universality theorem for periodic zeta-functions. II. Math. Modell. Analysis, 22(6):750–762, 2017. https://doi.org/10.3846/13926292.2017.1365779
R. Macaitienė, M. Stoncelis and D. Šiaučiūnas. A weighted universality theorem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017. https://doi.org/10.3846/13926292.2017.1269373
R. Macaitienė and D. Šiaučiūnas. Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function. Lith. Math. J., 59(1):81–95, 2019. https://doi.org/10.1007/s10986-019-09423-2
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7(2):31–122, 1952 (in Russian).
H. L. Montgomery. Topics in Multiplicative Number Theory. Lect. Notes Math., Vol. 227, Springer, Berlin, Heidelberg, New York, 1971. https://doi.org/10.1007/BFb0060851
H. L. Montgomery. The pair correlation of zeros of the zeta function. In H.G. Diamond(Ed.), Analytic Number Theory, volume 24 of Proc. Sympos. Pure Math., pp. 181–193. Amer. Math. Soc., Providence, RI, 1973. https://doi.org/10.1090/pspum/024/9944
J. Steuding. The roots of the equation ζ(s) = a are uniformly distributed modulo one. In A. Laurinčikas et al.(Ed.), Analytic and Probabilistic Methods in Number Theory, pp. 243–249, Vilnius, 2012. TEV.
E. C. Titchmarsh. The Theory of the Riemann zeta-function. 2nd ed. revised by D.R. Heath-Brown, Clarendon Press, Oxford, 1986.
V. Franckevič, A. Laurinčikas and D. Šiaučiūnas. On approximation of analytic functions by periodic Hurwitz zeta-functions. Math. Modell. Analysis, 24(1):20– 33, 2019. https://doi.org/10.3846/mma.2019.002
V. Garbaliauskienė, J. Karaliūnaitė and R. Macaitienė. On discrete value distribution of certain compositions. Math. Modell. Analysis, 24(1):34–42, 2019. https://doi.org/10.3846/mma.2019.003
R. Garunkštis and A. Laurinčikas. Discrete mean square of the Riemann zetafunction over imaginary parts of its zeros. Periodica Math. Hung., 76(2):217–228, 2018. https://doi.org/10.1007/s10998-017-0228-6
R. Garunkštis and A. Laurinčikas. The Riemann hypothesis and universality of the Riemann zeta-function. Math. Slovaca, 68(4):741–748, 2018. https://doi.org/10.1515/ms-2017-0141
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181(2):127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
A. Javtokas and A. Laurinčikas. Universality of the periodic Hurwitz zeta-function. Integral Transf. Spec. Functions, 17:711–722, 2006. https://doi.org/10.1080/10652460600856484
J. Kaczorowski. Some remarks on the universality of periodic L-functions. In R. Steuding and J. Steuding(Eds.), New directions in value-distribution theory of zeta and L-functions, pp. 113–120. Shaker Verlag, Aachen, 2009.
R. Kačinskaitė and A. Laurinčikas. The joint distribution of periodic zeta-functions. Studia Sci. Math. Hungarica, 48(2):257–279, 2011. https://doi.org/10.1556/SScMath.48.2011.2.1162
L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley and Sons, New York, 1974.
A. Laurinčikas. Voronin-type theorem for periodic Hurwitz zeta-functions. Sb. Math., 198:231–262, 2007. https://doi.org/10.1070/SM2007v198n02ABEH003835
A. Laurinčikas. Joint universality for periodic Hurwitz zeta-functions. Izv. Math., 72:741–760, 2008. https://doi.org/10.1070/IM2008v072n04ABEH002421
A. Laurinčikas. Joint universality of zeta-functions with periodic coefficients. Izv. Math., 74(3):515–539, 2010. https://doi.org/10.1070/IM2010v074n03ABEH002497
A. Laurinčikas. Universality of composite functions of periodic zeta-functions. Sb. Math., 203:1631–1643, 2012. https://doi.org/10.1070/SM2012v203n11ABEH004279
A. Laurinčikas. The joint discrete universality of periodic zeta-functions. In J. Sander et al.(Ed.), From Arithmetic to Zeta-Functions, Number Theory in Memory of Wolfgang Schwarz, pp. 231–246. Springer, 2016. https://doi.org/10.1007/978-3-319-28203-9¬_15
A. Laurinčikas. Universality theorems for zeta-functions with periodic coefficients. Sb. Math. J., 57:330–339, 2016. https://doi.org/10.1134/S0037446616020154
A. Laurinčikas. A discrete version of the Mishou theorem. II. Proc. Steklov Inst. Math., 296(1):172–182, 2017. https://doi.org/10.1134/S008154381701014X
A. Laurinčikas. On discrete universality of the Hurwitz zeta-functions. Results Math., 72(1-2):907–917, 2017. https://doi.org/10.1007/s00025-017-0702-8
A. Laurinčikas. Joint value distribution theorems for the Riemann and Hurwitz zeta-functions. Moscow Math. J., 18(2):349–366, 2018. https://doi.org/10.17323/1609-4514-2018-18-2-349-366
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. Quaest. Math., 42(5):687–699, 2019. https://doi.org/10.2989/16073606.2018.1481891
A. Laurinčikas. Non-trivial zeros of the Riemann zeta-function and joint universality theorems. J. Math. Anal. Appl., 475(1):395–402, 2019. https://doi.org/10.1016/j.jmaa.2019.02.047
A. Laurinčikas and R. Macaitienė. The discrete universality of the periodic Hurwitz zeta-function. Integral Transf. Spec. Functions, 20:673–686, 2009. https://doi.org/10.1080/10652460902742788
A. Laurinčikas and R. Macaitienė. Joint approximation of analytic functions by shifts of the Riemann and periodic Hurwitz zeta-functions. Appl. Anal. Discrete Math., 12(2):508–527, 2018. https://doi.org/10.2298/AADM170713016L
A. Laurinčikas, R. Macaitienė, D. Mochov and D. Šiaučiūnas. Universality of the periodic Hurwitz zeta-function with rational parameter. Siber. Math. J., 59(5):894–900, 2018. https://doi.org/10.1134/S0037446618050130
A. Laurinčikas and D. Mochov. A discrete universality theorem for the periodic Hurwitz zeta-functions. Chebysh. Sb., 17:148–159, 2016.
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210, 2019. https://doi.org/10.1070/SM9194
A. Laurinčikas and D. Šiaučiūnas. Remarks on the universality of periodic zeta-functions (in Russian). Math. Notes., 80(3-4):532–538, 2006. https://doi.org/10.1007/s11006-006-0171-y
R. Macaitienė, M.Stoncelis and Šiaučiūnas. A weighted discrete universality theorem for periodic zeta-functions. II. Math. Modell. Analysis, 22(6):750–762, 2017. https://doi.org/10.3846/13926292.2017.1365779
R. Macaitienė, M. Stoncelis and D. Šiaučiūnas. A weighted universality theorem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017. https://doi.org/10.3846/13926292.2017.1269373
R. Macaitienė and D. Šiaučiūnas. Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function. Lith. Math. J., 59(1):81–95, 2019. https://doi.org/10.1007/s10986-019-09423-2
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7(2):31–122, 1952 (in Russian).
H. L. Montgomery. Topics in Multiplicative Number Theory. Lect. Notes Math., Vol. 227, Springer, Berlin, Heidelberg, New York, 1971. https://doi.org/10.1007/BFb0060851
H. L. Montgomery. The pair correlation of zeros of the zeta function. In H.G. Diamond(Ed.), Analytic Number Theory, volume 24 of Proc. Sympos. Pure Math., pp. 181–193. Amer. Math. Soc., Providence, RI, 1973. https://doi.org/10.1090/pspum/024/9944
J. Steuding. The roots of the equation ζ(s) = a are uniformly distributed modulo one. In A. Laurinčikas et al.(Ed.), Analytic and Probabilistic Methods in Number Theory, pp. 243–249, Vilnius, 2012. TEV.
E. C. Titchmarsh. The Theory of the Riemann zeta-function. 2nd ed. revised by D.R. Heath-Brown, Clarendon Press, Oxford, 1986.