Some Results on L -Functions Related to Sharing Two Finite Sets
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Some Results on L-Functions Related to Sharing Two Finite Sets Pulak Sahoo1 · Samar Halder1 Received: 26 June 2018 / Revised: 1 February 2019 / Accepted: 21 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this article, we investigate the value distribution of L-functions in the (extended) Selberg class and establish two theorems which show how an L-function and a meromorphic function are uniquely determined by their sharing two finite sets. Our results answer a question of Lin and Lin (Filomat 30:3795–3806, 2016). Examples are also given in support of the accuracy of the results. Keywords L-function · Nevanlinna theory · Uniqueness · Shared set Mathematics Subject Classification Primary 30D35; Secondary 30D30 · 11M06 · 11M36
1 Introduction, Definitions, and Results The famous Riemann hypothesis states that all the non-trivial zeros of the Riemann 1 1 zeta function ζ (s) = ∞ n=1 n s lie on the critical line Re(s) = 2 of the complex plane C, where and throughout the paper, s denotes a complex variable. This has important implications in the distribution of prime numbers. Moreover, the Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar but more general, global L-functions. Naturally, L-functions with the Riemann zeta function as a prototype are important objects in number theory. Recently, the value distribution of L-functions has been studied extensively by many mathematicians
Communicated by Stephan Ruscheweyh. Samar Halder is supported by UGC-NFSC scheme of research fellowship.
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Pulak Sahoo [email protected] Samar Halder [email protected]
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Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India
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P. Sahoo, S. Halder
worldwide (see [3,5,7,12,15]). It concerns the distribution of zeros of L-functions and, more generally, the roots of the equation L(s) = c, where c is any value in the extended complex plane C = C ∪ {∞}. For a meromorphic function f and c ∈ C, let E(c, f ) denote the pre-image of c under f , given by E(c, f ) = {s ∈ C : f (s) − c = 0}, where each zero of f − c is counted with proper multiplicities. In addition, let E(c, f ) denote the set of distinct elements in E(c, f ). Two meromorphic functions f and g are said to share c CM (counting multiplicities) if E(c, f ) = E(c, g), and to share c IM (ignoring multiplicities) if E(c, f ) = E(c, g). The pre-image E(S, f ) of a set S(⊂ C) under f is defined by E(S, f ) =
{s ∈ C : f (s) − c = 0},
c∈S
where each zero of f −c is counted with proper multiplicities. In addition, by E(S, f ), we mean the set of distinct elements in E(S, f ). Two meromorphic functions f and g are said to share the set S CM, if E(S, f ) = E(S, g) and to share the set S IM, if E(S, f ) = E(S, g). The uniqueness problems regarding meromorphic functions and their shared values or sets have been extensively studied by many researchers (see [1,2,8,9,13,17]). This paper deals with shared value and shared set problems related to L-functions. In 19
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