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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 $$L(s,\chi)=c$$.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
$$L$$-functions; extended Selberg class; sharing values