Abstract

In this paper, we investigate the value distribution of L-functions in the extended Selberg class. We show how two L-functions $L_1$ and $L_2$ satisfying certain condition are uniquely determined by the zero sharing between $P(L_1)$ and $P(L_2)$ for some polynomial $P$, or by a set sharing between $L_1$ and $L_2$. Considering the most general form of a polynomial in the set sharing we obtain some results which completely generalize and extend some recent results of [X. M. Li, X. R. Du, H. X. Yi, Dirichlet series satisfying a Riemann type functional equation and sharing one set, Complex Var. Elliptic Equ., 68(10) (2023), 1653--1677], which were actually proved as an answer of an analogous question of Gross [F. Gross, Factorization of meromorphic functions and some open problems, Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, Ky., 1976), pp. 51--69, Lect. Notes Math., Vol 599, Springer, Berlin, 1977] for L-functions. We also obtain uniqueness relation between two nonconstant L-functions (belonging to the extended Selberg class) by proving other two results, one concerning a prior result due to Yuan-Li-Yi [Q. Q. Yuan, X. M. Li, H. X. Yi, Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58 (2018), 249--262] and another related to a result of Hao-Chen [W. J. Hao, J. F. Chen, Uniqueness theorems for L-functions in the extended Selberg class, Open Math., 16 (2018), 1291--1299].

Keywords: Dirichlet series; Selberg class; L-function; set sharing; uniqueness; meromorphic.

MSC: 30D35, 30D30, 11M06, 11M41

DOI: 10.57016/MV-OHTN3580

Pages:  1--13