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Uniqueness theorems for \(L\)-functions. (English) Zbl 1283.11128
The authors develop the Nevanlinna uniqueness theory [R. Nevanlinna, Eindeutige analytische Funktionen. 2. Aufl. Berlin etc.: Springer-Verlag (1953; Zbl 0050.30302)]; [C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1070.30011)]. They prove uniqueness theorems for \(L\)-functions from the extended Selberg class \(S^\#\).
Recall that two meromorphic functions \(f\) and \(g\) are said to share the value \(c\in \mathbb C\cup\{\infty\}\) counting multiplicities (CM) if the identity \(f^{-1}(c):=\{s \in \mathbb C: f(s)=c\}=g^{-1}(c)\) holds according to the same multiplicity of roots of the equations \(f(s)=c\) and \(g(s)=c\). If the multiplicities are ignored, then the functions \(f\) and \(g\) are said to share the vale \(c\) ignoring multiplicities (IM).
The authors prove that if the function \(f\) is meromorphic in \(\mathbb C\) and \({\mathcal L}\in S^\#\) is non-constant \(L\)-function (\(f\) and \({\mathcal L}\) share the values \(a,b \in \mathbb C\) CM and the value \(c \in \mathbb C\) IM), then \(f \equiv {\mathcal L}\).
Next result deals with a larger class of meromorphic functions, i.e., if the function \(f\) has finite non-zero order and one of conditions that the order of \(f\) is not integer or the order of \(f\) is an integer and \(f\) has maximal type, then \(f \equiv g\), when \(f\) and \(g\) are meromorphic non-constant functions in \(\mathbb C\) which share the values \(a,b\) CM and the value \(c\) IM and such that \(f\) or \(g\) assumes the value \(d\) only finitely many times (\(a, b,c,d \in \mathbb C\cup \{\infty\}\) are distinct).
Also, the authors generalize asymptotic formula for certain discrete moments of Dirichlet \(L\)-functions at the zeros of another Dirichlet \(L\)-function \(L(s,\chi)\) connecting to the roots of \(L(s,\chi)=c\).

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)