Discrete universality theorem for Matsumoto zeta-functions and nontrivial zeros of the Riemann zeta-function
Abstract
In 2017, Garunkštis, Laurinčikas and Macaitienė proved the discrete universality theorem for the Riemann zeta-function shifted by imaginary parts of nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended to various zeta-functions and L-functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions.
Keyword : Matsumoto zeta-function, universality, nontrivial zeros
How to Cite
Nakai, K. (2025). Discrete universality theorem for Matsumoto zeta-functions and nontrivial zeros of the Riemann zeta-function. Mathematical Modelling and Analysis, 30(1), 97–108. https://doi.org/10.3846/mma.2025.20817
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
B. Bagchi. Statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet series. PhD thesis, Indian Statistical InstituteKolkata, 1981.
A. Balčiūnas, V. Franckevič, V. Garbaliauskienė, R. Macaitienė and A. Rimkevičienė. Universality of zeta-functions of cusp forms and non-trivial zeros of the Riemann zeta-function. Math. Model. Anal., 26(1):82–93, 2021. https://doi.org/10.3846/mma.2021.12447
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. Model. Anal., 25(1):71–87, 2020. https://doi.org/10.3846/mma.2020.10450
K. Endo. Universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function. Lith. Math. J., 62:315–332, 2022. https://doi.org/10.1007/s10986-022-09568-7
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181:127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
R. Kačinskaitė. On discrete universality in the Selberg–Steuding class. Sib. Math. J., 63(2):277–285, 2022. https://doi.org/10.1134/S0037446622020069
R. Kačinskaitė and K. Matsumoto. Remarks on the mixed joint universality for a class of zeta functions. Bull. Aust. Math. Soc., 95(2):187–198, 2017. https://doi.org/10.1017/S0004972716000733
A. Klenke. Probability theory: a comprehensive course. Springer Science & Business Media, 2020. https://doi.org/10.1007/978-3-030-56402-5
E. Kowalski. An introduction to probabilistic number theory, volume 192. Cambridge University Press, 2021. https://doi.org/10.1017/9781108888226
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. Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function. Stud. Sci. Math. Hung., 57(2):147–164, 2020. https://doi.org/10.1556/012.2020.57.2.1460
A. Laurinčikas. Limit Theorems for the Riemann Zeta-function. Kluwer, 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. On the Matsumoto zeta-function. Acta Arith., 84:1–16, 1998. https://doi.org/10.4064/aa-84-1-1-16
A. Laurinčikas and K. Matsumoto. The universality of zeta-functions attached to certain cusp forms. Acta Arith., 98(4):345–359, 2002. https://doi.org/10.4064/aa98-4-2
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210(12):1753–1773, 2019. https://doi.org/10.1070/SM9194
K. Matsumoto. Value-distribution of zeta-functions. Lecture Notes in Math., 1434:178–187, 1990. https://doi.org/10.1007/BFb0097134
K. Matsumoto. A survey on the theory of universality for zeta and Lfunctions. Number Theory: Plowing and Starring Through High Wawe Forms, Proc. 7th China-Japan Semin.(Fukuoka 2013), 11:95–144, 2015. https://doi.org/10.1142/9789814644938_0004
H.L. Montgomery. Topics in Multiplicative Number Theory. Springer, 1971.
H.L. Montgomery. The pair correlation of zeros of the zeta function. In Proc. Symp. Pure Math, volume 24, pp. 181–193, 1973. https://doi.org/10.1007/BFb0060851
A. Reich. Werteverteilung von Zetafunktionen. Arch. Math., 34:440–451, 1980. https://doi.org/10.1007/BF01224983
A. Sourmelidis, T. Srichan and J. Steuding. On the vertical distribution of values of L-functions in the Selberg class. Int. J. Number Theory, 18(2):277–302, 2022. https://doi.org/10.1142/S1793042122500191
J. Steuding. Value-distribution of L-functions, volume 1877. Springer, 2007. https://doi.org/10.5565/PUBLMAT_PJTN05_12
S.M. Voronin. Theorem on the ”universality” of the Riemann zetafunction. Izv. Akad. Nauk SSSR Ser. Mat., 39(3):475–486, 1975. https://doi.org/10.1070/IM1975v009n03ABEH001485
A. Balčiūnas, V. Franckevič, V. Garbaliauskienė, R. Macaitienė and A. Rimkevičienė. Universality of zeta-functions of cusp forms and non-trivial zeros of the Riemann zeta-function. Math. Model. Anal., 26(1):82–93, 2021. https://doi.org/10.3846/mma.2021.12447
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. Model. Anal., 25(1):71–87, 2020. https://doi.org/10.3846/mma.2020.10450
K. Endo. Universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function. Lith. Math. J., 62:315–332, 2022. https://doi.org/10.1007/s10986-022-09568-7
R. Garunkštis, A. Laurinčikas and R. Macaitienė. Zeros of the Riemann zeta-function and its universality. Acta Arith., 181:127–142, 2017. https://doi.org/10.4064/aa8583-5-2017
R. Kačinskaitė. On discrete universality in the Selberg–Steuding class. Sib. Math. J., 63(2):277–285, 2022. https://doi.org/10.1134/S0037446622020069
R. Kačinskaitė and K. Matsumoto. Remarks on the mixed joint universality for a class of zeta functions. Bull. Aust. Math. Soc., 95(2):187–198, 2017. https://doi.org/10.1017/S0004972716000733
A. Klenke. Probability theory: a comprehensive course. Springer Science & Business Media, 2020. https://doi.org/10.1007/978-3-030-56402-5
E. Kowalski. An introduction to probabilistic number theory, volume 192. Cambridge University Press, 2021. https://doi.org/10.1017/9781108888226
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. Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function. Stud. Sci. Math. Hung., 57(2):147–164, 2020. https://doi.org/10.1556/012.2020.57.2.1460
A. Laurinčikas. Limit Theorems for the Riemann Zeta-function. Kluwer, 1996. https://doi.org/10.1007/978-94-017-2091-5
A. Laurinčikas. On the Matsumoto zeta-function. Acta Arith., 84:1–16, 1998. https://doi.org/10.4064/aa-84-1-1-16
A. Laurinčikas and K. Matsumoto. The universality of zeta-functions attached to certain cusp forms. Acta Arith., 98(4):345–359, 2002. https://doi.org/10.4064/aa98-4-2
A. Laurinčikas and J. Petuškinaitė. Universality of Dirichlet L-functions and non-trivial zeros of the Riemann zeta-function. Sb. Math., 210(12):1753–1773, 2019. https://doi.org/10.1070/SM9194
K. Matsumoto. Value-distribution of zeta-functions. Lecture Notes in Math., 1434:178–187, 1990. https://doi.org/10.1007/BFb0097134
K. Matsumoto. A survey on the theory of universality for zeta and Lfunctions. Number Theory: Plowing and Starring Through High Wawe Forms, Proc. 7th China-Japan Semin.(Fukuoka 2013), 11:95–144, 2015. https://doi.org/10.1142/9789814644938_0004
H.L. Montgomery. Topics in Multiplicative Number Theory. Springer, 1971.
H.L. Montgomery. The pair correlation of zeros of the zeta function. In Proc. Symp. Pure Math, volume 24, pp. 181–193, 1973. https://doi.org/10.1007/BFb0060851
A. Reich. Werteverteilung von Zetafunktionen. Arch. Math., 34:440–451, 1980. https://doi.org/10.1007/BF01224983
A. Sourmelidis, T. Srichan and J. Steuding. On the vertical distribution of values of L-functions in the Selberg class. Int. J. Number Theory, 18(2):277–302, 2022. https://doi.org/10.1142/S1793042122500191
J. Steuding. Value-distribution of L-functions, volume 1877. Springer, 2007. https://doi.org/10.5565/PUBLMAT_PJTN05_12
S.M. Voronin. Theorem on the ”universality” of the Riemann zetafunction. Izv. Akad. Nauk SSSR Ser. Mat., 39(3):475–486, 1975. https://doi.org/10.1070/IM1975v009n03ABEH001485