On mixed joint discrete universality for a class of zeta-functions: a further generalization
Abstract
We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.
Keyword : discrete shift, Matsumoto zeta-function, periodic Hurwitz zeta-function, simultaneous approximation, Steuding class, value distribution, weak convergence, universality
How to Cite
Kačinskaitė, R., & Matsumoto, K. (2020). On mixed joint discrete universality for a class of zeta-functions: a further generalization. Mathematical Modelling and Analysis, 25(4), 569-583. https://doi.org/10.3846/mma.2020.11751
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
P. Billingsley. Convergence of Probability Measures. Willey, New York, 1999.
E. Buivydas and A. Laurinčikas. A generalized joint discrete universality theorem for the Riemann and Hurwitz zeta-functions. Lith. Math. J., 55(2):193–206, 2015. https://doi.org/10.1007/s10986-015-9273-0
H. Cramér and M.R. Leadbetter. Stationary and Related Stochastic Processes. Wiley, New York, 1967.
J. Genys, R. Macaitienė, S. Račkauskienė and D. Šiaučiūnas. A mixed joint universality theorem for zeta-functions. Math. Modell. Anal., 15(4):431–446, 2010. https://doi.org/10.3846/1392-6292.2010.15.431-446
A. Javtokas and A. Laurinčikas. On the periodic Hurwitz zeta-function. Hardy-Ramanujan J., 29:18–36, 2006.
R. Kačinskaitė. A discrete limit theorem for the Matsumoto zeta-function on the complex plane. Lith. Math. J., 40(4):364–378, 2000. https://doi.org/10.1023/A:1007613613949
R. Kačinskaitė and B. Kazlauskaitė. Two results related to the universality of zeta-functions with periodic coeffcients. Results Math., 73(Article No. 95):73–95, 2018. https://doi.org/10.1007/s00025-018-0856-z
R. Kačinskaitė and A. Laurinčikas. The joint distribution of periodic zeta-functions. Studia Sci. Math. Hungar., 48(2):257–279, 2011. https://doi.org/10.1556/sscmath.48.2011.2.1162
R. Kačinskaitė and K. Matsumoto. The mixed joint universality for a class of zeta-functions. Math. Nachr., 288(16):1900–1909, 2015. https://doi.org/10.1002/mana.201400366
R. Kačinskaitė and K. Matsumoto. On mixed joint discrete universality for a class of zeta-functions. In A. Dubickas et al.(Ed.), Proc. of 6th Intern. Conf. Palanga, Anal. Probab. Methods in Number Theory, pp. 51–66, Vilnius, 2017. Vilnius University Publ. House.
R. Kačinskaitė and K. Matsumoto. Remarks on the mixed joint universality for a class of zeta-functions. Bull. Austral. Math. Soc., 98(2):187–198, 2017. https://doi.org/10.1017/S0004972716000733
R. Kačinskaitė and K. Matsumoto. On mixed joint discrete universality for a class of zeta-functions II. Lith. Math. J., 59(1):54–66, 2019. https://doi.org/10.1007/s10986-019-09432-1
A. Laurinčikas. Limit theorems for the Matsumoto zeta-function. J. Théor. Nombres Bordeaux, 8:143–158, 1996. https://doi.org/10.5802/jtnb.161
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer Academic Publisher, Dordrecht etc., 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. Voronin-type theorem for periodic Hurwitz zeta-functions. Sb. Math., 198:231–242, 2007. https://doi.org/10.1070/SM2007v198n02ABEH003835
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. A discrete universality theorem for the Hurwitz zeta-function. J. Number Theory, 143, 2014. https://doi.org/10.1016/j.jnt.2014.04.013
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. Quaest. Math., 42(5), 2019.
A. Laurinčikas. Joint discrete universality for periodic zeta-functions II. Quaest. Math., 2019. https://doi.org/10.2989/16073606.2019.1654554
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht etc., 2002.
A. Laurinčikas and R. Macaitienė. The discrete universality of the periodic Hurwitz zeta-function. Integr. Transf. Spec. Funct., 20:673–686, 2009. https://doi.org/10.1080/10652460902742788
A. Laurinčikas and S. Skerstonaitė. Joint universality for periodic Hurwitz zeta-functions II. In J. Steuding and R. Steuding(Eds.), Proc. of Conference, Würzburg, Germany, 2008, New Directions in Value-Distribution Theory of Zeta and L-Functions, pp. 149–159, Aachen, 2009. Shaker Verlag.
S.N. Mergelyan. Uniform approximations to functions of a complex variable. Am. Math. Soc. Transl., 101:99 p., 1954.
H. Mishou. The joint value-distribution of the Riemann zeta-function and Hurwitz zeta-functions. Lith. Math. J., 47(1):32–47, 2007. https://doi.org/10.1007/s10986-007-0003-0
J. Sander and J. Steuding. Joint universality for sums and products of Dirichlet L-functions. Analysis (Munich), 26(3):295–312, 2006. https://doi.org/10.1524/anly.2006.26.99.295
J. Steuding. Value-Distribution of L-Functions, volume 1877 of Lecture Notes Math. Springer Verlag, Berlin etc., 2007.
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Math. USSR Izv., 9:443–453, 1975. https://doi.org/10.1070/IM1975v009n03ABEH001485
E. Buivydas and A. Laurinčikas. A generalized joint discrete universality theorem for the Riemann and Hurwitz zeta-functions. Lith. Math. J., 55(2):193–206, 2015. https://doi.org/10.1007/s10986-015-9273-0
H. Cramér and M.R. Leadbetter. Stationary and Related Stochastic Processes. Wiley, New York, 1967.
J. Genys, R. Macaitienė, S. Račkauskienė and D. Šiaučiūnas. A mixed joint universality theorem for zeta-functions. Math. Modell. Anal., 15(4):431–446, 2010. https://doi.org/10.3846/1392-6292.2010.15.431-446
A. Javtokas and A. Laurinčikas. On the periodic Hurwitz zeta-function. Hardy-Ramanujan J., 29:18–36, 2006.
R. Kačinskaitė. A discrete limit theorem for the Matsumoto zeta-function on the complex plane. Lith. Math. J., 40(4):364–378, 2000. https://doi.org/10.1023/A:1007613613949
R. Kačinskaitė and B. Kazlauskaitė. Two results related to the universality of zeta-functions with periodic coeffcients. Results Math., 73(Article No. 95):73–95, 2018. https://doi.org/10.1007/s00025-018-0856-z
R. Kačinskaitė and A. Laurinčikas. The joint distribution of periodic zeta-functions. Studia Sci. Math. Hungar., 48(2):257–279, 2011. https://doi.org/10.1556/sscmath.48.2011.2.1162
R. Kačinskaitė and K. Matsumoto. The mixed joint universality for a class of zeta-functions. Math. Nachr., 288(16):1900–1909, 2015. https://doi.org/10.1002/mana.201400366
R. Kačinskaitė and K. Matsumoto. On mixed joint discrete universality for a class of zeta-functions. In A. Dubickas et al.(Ed.), Proc. of 6th Intern. Conf. Palanga, Anal. Probab. Methods in Number Theory, pp. 51–66, Vilnius, 2017. Vilnius University Publ. House.
R. Kačinskaitė and K. Matsumoto. Remarks on the mixed joint universality for a class of zeta-functions. Bull. Austral. Math. Soc., 98(2):187–198, 2017. https://doi.org/10.1017/S0004972716000733
R. Kačinskaitė and K. Matsumoto. On mixed joint discrete universality for a class of zeta-functions II. Lith. Math. J., 59(1):54–66, 2019. https://doi.org/10.1007/s10986-019-09432-1
A. Laurinčikas. Limit theorems for the Matsumoto zeta-function. J. Théor. Nombres Bordeaux, 8:143–158, 1996. https://doi.org/10.5802/jtnb.161
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer Academic Publisher, Dordrecht etc., 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. Voronin-type theorem for periodic Hurwitz zeta-functions. Sb. Math., 198:231–242, 2007. https://doi.org/10.1070/SM2007v198n02ABEH003835
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. A discrete universality theorem for the Hurwitz zeta-function. J. Number Theory, 143, 2014. https://doi.org/10.1016/j.jnt.2014.04.013
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. Quaest. Math., 42(5), 2019.
A. Laurinčikas. Joint discrete universality for periodic zeta-functions II. Quaest. Math., 2019. https://doi.org/10.2989/16073606.2019.1654554
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht etc., 2002.
A. Laurinčikas and R. Macaitienė. The discrete universality of the periodic Hurwitz zeta-function. Integr. Transf. Spec. Funct., 20:673–686, 2009. https://doi.org/10.1080/10652460902742788
A. Laurinčikas and S. Skerstonaitė. Joint universality for periodic Hurwitz zeta-functions II. In J. Steuding and R. Steuding(Eds.), Proc. of Conference, Würzburg, Germany, 2008, New Directions in Value-Distribution Theory of Zeta and L-Functions, pp. 149–159, Aachen, 2009. Shaker Verlag.
S.N. Mergelyan. Uniform approximations to functions of a complex variable. Am. Math. Soc. Transl., 101:99 p., 1954.
H. Mishou. The joint value-distribution of the Riemann zeta-function and Hurwitz zeta-functions. Lith. Math. J., 47(1):32–47, 2007. https://doi.org/10.1007/s10986-007-0003-0
J. Sander and J. Steuding. Joint universality for sums and products of Dirichlet L-functions. Analysis (Munich), 26(3):295–312, 2006. https://doi.org/10.1524/anly.2006.26.99.295
J. Steuding. Value-Distribution of L-Functions, volume 1877 of Lecture Notes Math. Springer Verlag, Berlin etc., 2007.
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Math. USSR Izv., 9:443–453, 1975. https://doi.org/10.1070/IM1975v009n03ABEH001485