Share:


On the Functional Independence of the Riemann Zeta-Function

    Virginija Garbaliauskienė   Affiliation
    ; Renata Macaitienė   Affiliation
    ; Darius Šiaučiūnas   Affiliation

Abstract

In 1973, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i.e., that ζ(s) and its derivatives do not satisfy a certain equation with continuous functions. In the paper, we obtain a joint version of the Voronin theorem.

Keyword : functional independence, Riemann zeta-function, universality of zeta-functions

How to Cite
Garbaliauskienė, V., Macaitienė, R., & Šiaučiūnas, D. (2023). On the Functional Independence of the Riemann Zeta-Function. Mathematical Modelling and Analysis, 28(2), 352–359. https://doi.org/10.3846/mma.2023.17157
Published in Issue
Mar 21, 2023
Abstract Views
313
PDF Downloads
435
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

B. Bagchi. The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series. PhD Thesis, Indian Statistical Institute, Calcutta, 1981.

H. Bohr and R. Courant. Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion. Reine Angew. Math., 144:249–274, 1914. https://doi.org/10.1515/crll.1914.144.249

R. Garunkštis and A. Laurinčikas. The Lerch zeta-function. Integral Transforms Spec. Funct., 10(3–4):211–226, 2000. https://doi.org/10.1080/10652460008819287

O. Hölder. Über die Eigenschaft der Gammafunktion keiner algebraischen Differentialgleichung zu genügen. Math. Ann., 28:1–13, 1887. https://doi.org/10.1007/BF02430507

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. Hungar., 48(2):257–279, 2011. https://doi.org/10.1556/sscmath.48.2011.2.1162

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

J. Kaczorowski, A. Laurinčikas and J. Steuding. On the value distribution of shifts of universal Dirichlet series. Monatsh. Math., 147(4):309–317, 2006. https://doi.org/10.1007/s00605-005-0339-7

A.A. Karatsuba and S.M. Voronin. The Riemann Zeta-Function. Walter de Gruyter, Berlin, 1992. https://doi.org/10.1515/9783110886146

A. Laurinčikas. Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996. https://doi.org/10.1007/978-94-017-2091-5

A. Laurinčikas. Functional independence of periodic Hurwitz zeta functions. Math. Notes, 83(1–2):65–71, 2008. https://doi.org/10.1134/S0001434608010082

A. Laurinčikas. Extension of the functional independence of the Riemann zetafunction. Glas. Mat., 55(1):55–65, 2020. https://doi.org/10.3336/gm.55.1.05

A. Laurinčikas. On a generalization of Voronin’s theorem. Math. Notes, 107(3– 4):442–451, 2020. https://doi.org/10.1134/S0001434620030086

A. Laurinčikas. On joint universality of the Riemann zeta-function. Math. Notes, 110(1–2):210–220, 2021. https://doi.org/10.1134/S0001434621070221

A. Laurinčikas and K. Matsumoto. The joint universality and the functional independence for Lerch zeta-functions. Nagoya Math. J., 157:211–227, 2000. https://doi.org/10.1017/S002776300000725X

A. Laurinčikas and K. Matsumoto. The joint universality of twisted automorphic L-functions. J. Math. Soc. Japan, 56(3):923–939, 2004. https://doi.org/10.2969/jmsj/1191334092

A. Laurinčikas, K. Matsumoto and J. Steuding. The universality of L-functions associated with new forms. Izv. Math., 67(1):77–90, 2003. https://doi.org/10.1070/IM2003v067n01ABEH000419

A. Laurinčikas, K. Matsumoto and J. Steuding. Discrete universality of L-functions of new forms. II. Lith. Math. J., 56(2):207–218, 2016. https://doi.org/10.1007/s10986-016-9314-3

F. Liu. A remark on the regularity of the discrete maximal operators. Bull. Aust. Math. Soc., 95:108–120, 2017. https://doi.org/10.1017/S0004972716000940

K. Matsumoto. A survey on the theory of universality for zeta and L-functions. In M. Kaneko, S. Kanemitsu and J. Liu(Eds.), Number Theory: Plowing and Starring Through High Wawe Forms, Proc. 7th China-Japan Semin. (Fukuoka 2013), volume 11 of Number Theory and Appl., pp. 95–144, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2015. World Scientific Publishing Co. https://doi.org/10.1142/9789814644938_0004

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

H. Mishou. The joint value distribution of the Riemann zeta function and Hurwitz zeta functions II. Arch. Math., 90(3):230–238, 2008. https://doi.org/10.1007/s00013-007-2397-7

H. Mishou. Joint universality theorem of Selberg zeta functions for principal congruence subgroups. J. Number Th., 227(3):235–264, 2021. https://doi.org/10.1016/j.jnt.2021.03.009

D.D. Mordukhai-Boltovskoi. On the Hilbert problem. Izv. Politech. Inst. Warszawa, 1914 (in Russian).

H. Nagoshi. Hypertranscendence of L-functions for GL(m)(A(Q)). Bull. Aust. Math. Soc., 93(3):388–399, 2016. https://doi.org/10.1017/S000497271500129X

H. Nagoshi. On a certain set of Lerch’s zeta-functions and their derivatives. Lith. Math. J., 59(1):111–130, 2019. https://doi.org/10.1007/s10986-019-09433-0

T. Nakamura. Zeros and the universality for the Euler–Zagier–Hurwitz type of multiple zeta-functions. Bull. Lond. Math. Soc., 41(4):691–700, 2009. https://doi.org/10.1112/blms/bdp043

A. Ostrowski. Über Dirichletsche Reihen und algebraische Differentialgleichungen. Math. Z., 8:241–298, 1920. https://doi.org/10.1007/BF01206530

A.G. Postnikov. A generalization of one Hilbert’s problem. DAN SSSR, 107(4):512–515, 1956 (in Russian).

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. On the distribution of nonzero values of the Riemann ζ-function. Trudy Mat. Inst. Steklov, 128:131–150, 1972 (in Russian).

S.M. Voronin. The differential independence of ζ-functions. DAN SSSR, 209:1264–1266, 1973 (in Russian).

S.M. Voronin. The functional independence of Dirichlet L-functions. Acta Arith., 27:493–503, 1975 (in Russian). https://doi.org/10.4064/aa-27-1-493-503

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. Selected works: Mathematics. (ed. A.A. Karatsuba), Publishing House MGTU Im. N.E. Baumana, Moscow, 2006 (in Russian).