A weighted version of the Mishou theorem
Abstract
In 2007, H. Mishou obtained a joint universality theorem for the Riemann and Hurwitz zeta-functions ζ(s) and ζ(s,α) with transcendental parameter α on the approximation of a pair of analytic functions by shifts (ζ(s+iτ),ζ(s+iτ,α)), τ ∈R. In the paper, the Mishou theorem is generalized for the set of above shifts having a weighted positive lower density. Also, the case of a positive density is considered.
Keyword : Hurwitz zeta-function, Mishou theorem, Riemann zeta-function, universality
How to Cite
Laurinčikas, A., Šiaučiūnas, D., & Vadeikis, G. . (2021). A weighted version of the Mishou theorem. Mathematical Modelling and Analysis, 26(1), 21-33. https://doi.org/10.3846/mma.2021.12445
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Laurinčikas. The joint universality of Hurwitz zeta-functions. Šiauliai Math. Semin., 3(11):169–187, 2008.
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 theorems for zeta-functions with periodic coefficients. Siber. Math. J., 57(2):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. 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):385–402, 2019. https://doi.org/10.1016/j.jmaa.2019.02.047
A. Laurinčikas. On the Mishou theorem with algebraic parameter. Siber. Math. J., 60(6):1075–1082, 2019. https://doi.org/10.1134/S0037446619060144
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. II. Quaest. Math., 2020. https://doi.org/10.2989/16073606.2019.1654554
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht, Boston, London, 2002. https://doi.org/10.1007/978-94-017-6401-8
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, D. Šiaučiūnas and G. Vadeikis. Weighted discrete universality of the Riemann zeta-function. Math. Modell. Anal., 25(1):21–36, 2020. https://doi.org/10.3846/mma.2020.10436
R. Macaitienė, M. Stoncelis and D. 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 the-orem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017. https://doi.org/10.3846/13926292.2017.1269373
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7:31–122, 1952 (in Russian).
H. Mishou. The joint value distribution of the Rieman zeta-function and Hurwitz zeta-functions. Lith. Math. J., 47(1):32–47, 2007. https://doi.org/10.1007/s10986-007-0003-0
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math. vol. 1877, Springer, Berlin, Heidelberg, 2007. https://doi.org/10.5565/PUBLMAT_PJTN05_12
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Izv. Akad. Nauk SSSR, Ser. Matem., 39:475–486, 1975 (in Russian).
S.M. Voronin. Analytic Properties of Generating Function of Arithmetic Objects. Diss. doktor fiz.-matem. nauk, Steklov Math. Inst., Moscow, 1977 (in Russian).
A. Balčiūnas, V. Garbaliauskienė, J. Karaliūnaitė, R. Macaitienė, J. Petuškinaitė and A. Rimkevičienė. Joint discrete approximation of a pair of analytic functions by periodic zeta-functions. Math. Modell. Anal., 25(1):71–87, 2020. https://doi.org/10.3846/mma.2020.10450
A. Balčiūnas and G. Vadeikis. A weighted universality throrem for the Hurwitz zeta-function. Šiauliai Math. Semin. , 12(20):5–18, 2017.
P. Billingsley. Convergence of Probability Measures. Willey, New York, 1968.
V. Garbaliauskienė, J. Karaliūnaitė and A. Laurinčikas. On zeros of some combinations of Dirichlet L-functions and Hurwitz zeta-functions. Math. Modell. Anal., 22(6):733–749, 2017. https://doi.org/10.3846/13926292.2017.1365313
S.M. Gonek. Analytic Properties of Zeta and L-Functions. PhD Thesis, University of Michigan, 1979.
R. Kačinskaitė and B. Kazlauskaitė. Two remarks related to the universality of zeta-functions with periodic coefficients. Results Math., 73(3):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. Hungarica, 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. Remarks on the mixed joint universality for a class of zeta-functions. Bull. Austral. Math. Soc., 95(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. On the universality of the Riemann zeta-function. Lith. Math. J., 35(4):399–402, 1995. https://doi.org/10.1007/BF02348827
A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer Academic Publishers, Dordrecht, Boston, London, 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. The joint universality of Hurwitz zeta-functions. Šiauliai Math. Semin., 3(11):169–187, 2008.
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 theorems for zeta-functions with periodic coefficients. Siber. Math. J., 57(2):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. 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):385–402, 2019. https://doi.org/10.1016/j.jmaa.2019.02.047
A. Laurinčikas. On the Mishou theorem with algebraic parameter. Siber. Math. J., 60(6):1075–1082, 2019. https://doi.org/10.1134/S0037446619060144
A. Laurinčikas. Joint discrete universality for periodic zeta-functions. II. Quaest. Math., 2020. https://doi.org/10.2989/16073606.2019.1654554
A. Laurinčikas and R. Garunkštis. The Lerch Zeta-Function. Kluwer Academic Publishers, Dordrecht, Boston, London, 2002. https://doi.org/10.1007/978-94-017-6401-8
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, D. Šiaučiūnas and G. Vadeikis. Weighted discrete universality of the Riemann zeta-function. Math. Modell. Anal., 25(1):21–36, 2020. https://doi.org/10.3846/mma.2020.10436
R. Macaitienė, M. Stoncelis and D. 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 the-orem for periodic zeta-functions. Math. Modell. Analysis, 22(1):95–105, 2017. https://doi.org/10.3846/13926292.2017.1269373
S.N. Mergelyan. Uniform approximations to functions of complex variable. Usp. Mat. Nauk., 7:31–122, 1952 (in Russian).
H. Mishou. The joint value distribution of the Rieman zeta-function and Hurwitz zeta-functions. Lith. Math. J., 47(1):32–47, 2007. https://doi.org/10.1007/s10986-007-0003-0
J. Steuding. Value-Distribution of L-Functions. Lecture Notes Math. vol. 1877, Springer, Berlin, Heidelberg, 2007. https://doi.org/10.5565/PUBLMAT_PJTN05_12
S.M. Voronin. Theorem on the “universality” of the Riemann zeta-function. Izv. Akad. Nauk SSSR, Ser. Matem., 39:475–486, 1975 (in Russian).
S.M. Voronin. Analytic Properties of Generating Function of Arithmetic Objects. Diss. doktor fiz.-matem. nauk, Steklov Math. Inst., Moscow, 1977 (in Russian).