On the Functional Independence of the Riemann Zeta-Function
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
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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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).
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).