On a result of Bao-Qin Li concerning Dirichlet series and shared values

Xiao-Yan Fan¹ ; Xiao-Min Li¹ ; Hong-Xun Yi²

¹ Ocean University of China, Department of Mathematics
² Shandong University, Department of Mathematics

Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 779-795.

Résumé

Li (2011) proved that if two L-functions $L_1$ and $L_2$ in the extended Selberg class ${\mathcal{S}}^{\sharp}$ satisfy the same functional equation with $a(1)=1$ and $L_1^{-1}(c_{j})=L_2^{-1}(c_{j})$ with $j\in\{1,2\}$ for two distinct finite complex numbers $c_{1}$ and $c_{2}$, then $L_{1}=L_{2}$. Later on, Gonek--Haan--Ki (2014) proved that if two L-functions $L_1$ and $L_2$ in the extended Selberg class ${\mathcal {S}}^{\sharp}$ have the positive degrees and $L_1^{-1}(c)=L_2^{-1}(c)$ for a finite non-zero complex number $c$, then $L_1=L_2$. This implies that if two L-functions $L_1$ and $L_2$ in the extended Selberg class ${\mathcal {S}}^{\sharp}$ have the positive degrees and $L_1^{-1}(c_j)=L_2^{-1}(c_j)$ with $j\in\{1,2\}$ for two distinct finite complex numbers $c_1$ and $c_2$, then $L_1=L_2$. In this paper, we prove that if two L-functions $L_1$ and $L_2$ in the extended Selberg class ${\mathcal{S}}^{\sharp}$ have the zero degrees and satisfy $L_1^{-1}(c_{j})=L_2^{-1}(c_{j})$ with $j\in\{1,2\}$ for two distinct finite complex numbers $c_{1}$ and $c_{2}$, and if $a_1(1)=a_2(1)$ or $\lim_{r\rightarrow+\infty}\frac{T(r,L_2)}{T(r,L_1)}=1$, then $L_{1}= L_{2}$. The main results obtained in this paper improve Theorem 1 from Li (2011) when the L-functions in the extended Selberg class ${\mathcal {S}}^{\sharp}$ have the zero degrees. Some examples are provided to show that the results obtained in this paper, in a sense, are best possible.

Détail

DOI : 10.54330/afm.141539

Keywords: Nevanlinna theory, meromorphic functions, Selberg class, extended Selberg class, shared values, uniqueness of L-functions