A Result on Value Distribution of L-functions

We will establish a theorem on value distribution of L-functions in the Selberg class, which shows how an L-function and a meromorphic function are uniquely determined by their c-values and which, as a consequence, proves a result on the unicity of the Riemann zeta function. L-functions, with the Riemann zeta function as a prototype, are important objects in number theory, and value distribution of L-functions has been studied extensively (cf. the recent monograph [St] and various references therein). Value distribution of L-functions concerns distribution of the zeros of L-functions L and, more generally, the c-values of L, i.e., the roots of the equation L(s) = c, or the values in the preimage L−1(c) = {s ∈ C : L(s) = c}, where and throughout the paper, s denotes the complex variable in the complex plane C and c denotes a complex value. L-functions can be analytically continued as meromorphic functions in C. It is well-known that a nonconstant meromorphic function in C is completely determined by five such preimages (see e.g. [H]), which is a famous theorem due to Nevanlinna and often referred to as Nevanlinna’s uniqueness or unicity theorem. Two meromorphic functions f and g in the complex plane are said to share a value c ∈ C∪{∞} IM (ignoring multiplicities) if f−1(c) = g−1(c) as two sets in C. Moreover, f and g are said to share a value c CM (counting multiplicities) if they share the value c and if the roots of the equations f(s) = c and g(s) = c have the same multiplicities. In terms of sharing values, two nonconstant meromorphic functions in C must be identically equal if they share five values IM, and one must be a Möbius transform of the other if they share four values CM; the numbers “five” and “four” are the best possible, as shown by Nevanlinna (see e.g. [H] or [St]). This paper concerns the question of how an L-function is uniquely determined in terms of the preimages of complex values, or sharing values. We refer the reader to the monograph [St] for a detailed discussion on the topic and related works. Throughout the paper, an L-function always means an L-function L in the Selberg class, which includes the Riemann zeta function ζ(s) = ∑∞ n=1 n −s and essentially those Dirichlet series where one might expect a Riemann hypothesis. Such an Lfunction is defined to be a Dirichlet series L(s) = ∑∞ n=1 a(n)n −s, satisfying the following axioms ([St] and [Se]): (i) Ramanujan hypothesis. a(n) n for every ε > 0. (ii) Analytic continuation. There is a non-negative integer k such that (s− 1)L(s) is an entire function of finite order. Received by the editors June 13, 2009, and, in revised form, September 20, 2009. 2010 Mathematics Subject Classification. Primary 30D30, 30D35, 11M06, 11M36. c ©2009 American Mathematical Society Reverts to public domain 28 years from publication 2071 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use