Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Value Distribution of L-functions and a Question of Chung-Chun Yang

  • Li, Xiao-Min (Department of Mathematics, Ocean University of China) ;
  • Yuan, Qian-Qian (Department of Mathematics, Ocean University of China) ;
  • Yi, Hong-Xun (Department of Mathematics, Shandong University)
  • Received : 2017.02.22
  • Accepted : 2021.05.18
  • Published : 2021.09.30
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Abstract

We study the value distribution theory of L-functions and completely resolve a question from Yang [10]. This question is related to L-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from Li [10].

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Acknowledgement

The authors wish to express their thanks to the referee for his/her valuable suggestions and comments.

References

  1. E. Bombieri and A. Perelli, Distinct zeros of L-functions, Acta Arith., 83(3)(1998), 271-281. https://doi.org/10.4064/aa-83-3-271-281
  2. G. G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc., s2-20(3)(1979), 457-466. https://doi.org/10.1112/jlms/s2-20.3.457
  3. G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc., s2-37(1)(1988), 88-104. https://doi.org/10.1112/jlms/s2-37.121.88
  4. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  5. W. K. Hayman and J. Miles, On the growth of a meromorphic function and its derivatives, Complex Var. Theory Appl., 12(1-4)(1989), 245-260. https://doi.org/10.1080/17476938908814369
  6. P. C. Hu and B. Q. Li, A simple proof and strengthening of a uniqueness theorem for L-functions, Canadian math. Bull., 59(1)(2016), 119-122. https://doi.org/10.4153/CMB-2015-045-1
  7. H. Ki, A remark on the uniqueness of the Dirichlet series with a Riemann-type function equation, Adv. Math. 231(5)(2012), 2484-2490. https://doi.org/10.1016/j.aim.2012.07.027
  8. I. Lahiri and A. Sarkar, On uniqueness theorem of Tohge, Arch. Math., 84(5)(2005), 461-469. https://doi.org/10.1007/s00013-004-1217-6
  9. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  10. B. Q. Li, A result on value distribution of L-functions, Proc. Amer. Math. Soc., 138(6)(2009), 2071-2077. https://doi.org/10.1090/S0002-9939-09-10222-8
  11. B. Q. Li , A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation, Adv. Math., 226(5)(2011), 4198-4211. https://doi.org/10.1016/j.aim.2010.12.001
  12. B. Q. Li, On common zeros of L-functions, Math. Z., 272(3-4)(2012), 1097-1102.
  13. W.C. Lin and K. Tohge, On shared-value properties of Painlev'e Transcendents, Comput. methods Funct.theory, 7(2)(2007), 477-499. https://doi.org/10.1007/BF03321658
  14. A. Z. Mokhon-ko, On the Nevanlinna characteristics of some meromorphic functions, in: Theory of Functions, Functional Analysis and Their Applications, vol.14, Izd-vo Khar'kovsk. Un-ta, 1971, pp. 83-87.
  15. R. Nevanlinna, Einige Eindeutigkeitssatze in der Theorie der meromorphen Funktionen, Acta Math., 48(3)(1926), 367-391. https://doi.org/10.1007/BF02565342
  16. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), E. Bombieri et al. (eds.), Collected papers, Vol. II, Springer-Verlag, 1991, pp. 47-63.
  17. J. Steuding, Value distribution of L-functions, Lecture Notes in Mathematics, 1877, Springer-Verlag, Berlin, 2007.
  18. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2003.
  19. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin Heidelberg, 1993.
  20. H. X. Yi, Unicity theorems for meromorphic functions that share three values, Kodai Math. J., 18(2)(1995), 300-314. https://doi.org/10.2996/kmj/1138043427
  21. Q. C. Zhang, Meromorphic functions sharing three values, Indian J. Pure Appl. Math., 30(7)(1999), 667-682.
  22. Q. C. Zhang, Meromorphic functions sharing values or sets, Doctoral Thesis, Shandong University, 1999.