Some uniqueness results related to \(L\)-functions. (English) Zbl 1409.30028
Author’s abstract: In this paper, we describe a family of meromorphic functions in C from analyzing some properties of these \(L\)-functions in the extended Selberg class and show two uniqueness results of such a function, in terms of shared values with a general meromorphic function in C. In particular, we show the condition “1CM+3IM value-sharing” suffices.
Reviewer: Zehua Zhou (Tianjin)
MSC:
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
References:
| [1] | Adams, W., Straus, E.: Non-archimedian analytic functions taking the same values at the same points. Illinois J. Math. 15, 418-424 (1971) · Zbl 0215.13202 |
| [2] | Garunkštis, R., Grahl, J., Steuding, J.: Uniqueness theorems for \[L\] L-functions. Comment. Math. Univ. St. Pauli 60, 15-35 (2011) · Zbl 1283.11128 |
| [3] | Gonek, S., Haan, J., Ki, H.: A uniqueness theorem for functions in the extended Selberg class. Math. Z. 278, 995-1004 (2014) · Zbl 1306.11069 · doi:10.1007/s00209-014-1343-1 |
| [4] | Gundersen, G.G.: Meromorphic functions that share three values IM and a fourth value CM. Complex Variables Theory Appl. 20, 99-106 (1992) · Zbl 0773.30032 · doi:10.1080/17476939208814590 |
| [5] | Hayman, W.K.: Meromorphic functions. Oxford University Press, Oxford (1964) · Zbl 0115.06203 |
| [6] | Hu, P., Li, B.Q.: A simple proof and strengthening of a uniqueness theorem for \[L\] L-functions. Canad. Math. Bull. 59, 119-122 (2016) · Zbl 1334.30001 · doi:10.4153/CMB-2015-045-1 |
| [7] | Li, B.Q.: A result on value distribution of \[L\] L-functions. Proc. Amer. Math. Soc. 138, 2071-2077 (2010) · Zbl 1195.30041 · doi:10.1090/S0002-9939-09-10222-8 |
| [8] | Li, X., Yi, H.: Results on value distribution of \[L\] L-functions. Math. Nachr. 286, 1326-1336 (2013) · Zbl 1291.30185 · doi:10.1002/mana.201200196 |
| [9] | Mues, E.: Meromorphic functions sharing four values. Complex Variables Theory Appl. 12, 169-179 (1989) · Zbl 0699.30025 · doi:10.1080/17476938908814363 |
| [10] | Nevanlinna, R.: Einige Eindeutigkeitssätze in der Theorie der Meromorphen Funktionen. Acta Math. 48, 367-391 (1926) · JFM 52.0323.03 · doi:10.1007/BF02565342 |
| [11] | Nevanlinna, R.: Analytic functions. Springer-Verlag, New York-Berlin (1970) · Zbl 0199.12501 · doi:10.1007/978-3-642-85590-0 |
| [12] | Ozawa, M.: Unicity theorems for entire functions. J. Analyse Math. 30, 411-420 (1976) · Zbl 0337.30020 · doi:10.1007/BF02786728 |
| [13] | Selberg, A.: Old and new conjectures and results about a class of Dirichlet series. Proceedings of the Amalfi conference on analytic number theory (Maiori, 1989), 367-385. University of Salerno, Salerno, (1992) · Zbl 0787.11037 |
| [14] | Steinmetz, N.: Reminiscence of an open problem: remarks on Nevanlinna’s four-value-theorem. Southeast Asian Bull. Math. 36, 399-417 (2012) · Zbl 1274.30127 |
| [15] | Steuding, J.: Value-distribution of \[L\] L-functions. Springer, Berlin (2007) · Zbl 1130.11044 |
| [16] | Ueda, H.: Some estimates for meromorphic functions sharing four values. Kodai Math. J. 17, 329-340 (1994) · Zbl 0820.30021 · doi:10.2996/kmj/1138039971 |
| [17] | Yang, C.C., Yi, H.: Uniqueness theory of meromorphic functions. Kluwer Academic Publishers, Dordrecht (2003) · Zbl 1070.30011 · doi:10.1007/978-94-017-3626-8 |
| [18] | Yang, L.: Value distribution theory. Springer-Verlag, Berlin (1993) · Zbl 0790.30018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
