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A Connection Between the Riemann Hypothesis and Uniqueness

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A Connection Between the Riemann Hypothesis and Uniqueness A Connection between the Riemann Hypothesis and Uniqueness of the Riemann zeta function Pei-Chu Hu and Bao Qin Li Department of Mathematics, Shandong University, Jinan 250100, Shandong, P. R. China E-mail: [email protected] Department of Mathematics and Statistics, Florida International University, Miami, FL 33199 USA E-mail: libaoqin@fiu.edu Abstract In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L- functions. 1 Introduction The Riemann ζ function is defined by the Dirichlet series ∞ arXiv:1610.01583v1 [math.NT] 5 Oct 2016 1 ζ(s)= , s = σ + it (1.1) ns nX=1 Mathematics Subject Classification 2000 (MSC2000): 11M36, 30D35. Key words and phrases: Riemann zeta function; Riemann hypothesis, uniqueness, the Dedekind zeta function, L-function, Riemann functional equation. 1 for Re(s) > 1, which is absolutely convergent, and admits an analytical continuation as a meromorphic function in the complex plane C of order 1, which has only a simple pole at s = 1 with residue equal to 1. It satisfies the following Riemann functional equation: πs ζ(1 s) = 2(2π)−s cos Γ(s)ζ(s), (1.2) − 2 where Γ is the Euler gamma function ∞ Γ(z)= tz−1e−tdt, Rez > 0, Z0 analytically continued as a meromorphic function in C of order 1 without any zeros and with simple poles at s = 0, 1, 2, . − − · · · The allied function s s s ξ(s)= (s 1)π− 2 Γ ζ(s) (1.3) 2 − 2 is an entire function of order equal to 1 satisfying the functional equation ξ(1 s)= ξ(s) (1.4) − (see e.g. [16], p.16 and p.29). It is easy to see that ζ(s) has no zeros for Re(s) > 1 and, by the functional equation, the only zeros of ζ(s) in the domain Re(s) < 0 are the poles of Γ(s/2). These are called the trivial zeros of ζ(s). Other zeros, called nontrivial zeros, lie in the critical strip 0 Re(s) 1 (actually lie in the ≤ ≤ open strip 0 < Re(s) < 1). It is a well-known theorem of G. H. Hardy that 1 there are an infinity of zeros on Re(s) = 2 . The famous, as yet unproven, Riemann hypothesis states as follows: Conjecture 1.1 (Riemann Hypothesis). The nontrivial zeros of ζ(s) lie on 1 the line Re(s)= 2 . The uniqueness problem for the Riemann zeta function (more generally, for L-functions, see below) is to study how the Riemann zeta function ζ (or an L-function) is uniquely determined by its zeros or by its a-values, i.e., the zeros of ζ(s) a, where a is a complex value. Uniqueness problems have − extensively been studied in the value distribution theory of meromorphic functions in terms of shared values (see e.g. the monographs [5] and [17]), in which two meromorphic functions f and g are called to share a value a if Z(f a)= Z(g a), where Z(F ) denotes the zero set of F (counting or not − − 2 counting multiplicities, depending on the questions under consideration). The problem for the Riemann zeta function and L-functions has recently been studied in various settings (see e.g. [2], [3], [7], [8], [10], [11], [14], to list a few). In particular, in [8] and [10], the problem was considered by relaxing the set equality Z(f a)= Z(g a) to the set inclusion Z(f a) − − − ⊆ Z(g a) for uniqueness of L-functions, which will be seen to be crucial in 2. − § Roughly speaking, two L-functions satisfying the same functional equation are identically equal if they have sufficiently many common zeros (see [8] and [10] for the details and related results as well as references), which gives a uniqueness theorem for solutions of the Riemann functional equation or, more generally, Riemann type functional equations (cf. 2 and see [1], [4], § [9], etc. for studies of solutions of the Riemann functional equation). In the present paper, we will discover a connection (an equivalence) be- tween the Riemann Hypothesis and the above mentioned uniqueness prob- lem and then an analogue for L-functions, which, as a consequence, also implies a simply stated necessary and sufficient condition for the Riemann Hypothesis to hold in terms of the limit of an allied function as σ + . → ∞ This connection does not seem to have been observed before. The results it has brought out in this paper are of a neat and best possible form. Given the fact that uniqueness problems have been studied extensively for meromor- phic functions and various techniques have been developed over the years, it would be profitable to further explore this approach with the connection in mind. 2 Results Let ρ be the nontrivial zeros of ζ in the critical strip 0 Re(s) 1. It n ≤ ≤ follows that ζ(ρ )= ζ(1 ρ )= ζ(¯ρ )= ζ(1 ρ¯ ) = 0 n − n n − n from the functional equation and the identity ζ(¯s)= ζ(s), that is,ρ ¯ , 1 ρ , n − n 1 ρ¯ are zeros of ζ(s), too. In other words, nontrivial zeros of ζ(s) are − n distributed symmetrically with respect to the real axis and to the critical 1 line Re(s)= 2 . 1 Now, let sν be the zeros of ζ on the half-line Re(s) = 2 , Im(s) > 0. Assume that ρn, sν are ordered with respect to increasing absolute values of their imaginary parts. We would like to define a meromorphic function that captures the key features of the Riemann zeta function ζ with, however, all its zeros in the critical strip being located exactly at those nontrivial zeros of ζ on the 3 1 critical line Re(s)= 2 . If this function is defined “ideally” and turns out to be identically equal to ζ (It is here where the uniqueness problem arises, cf. below), then the Riemann hypothesis must follow by the distribution of the zeros of the constructed function. To realize this goal, we first construct an entire function which plays the role of ξ with, however, zeros at sν, which are, as mentioned above, distributed symmetrically with respect to the critical line. We define ∞ 1 s s2 h(s)= 1 − . (2.1) 2 − s 2 νY=1 | ν| This function h possesses the following properties, which are important in serving our purposes: (a) h is an entire function of order 1; s−s2 ≤ (b) The general factor 1 2 in the infinite product has exactly the − |sν | zeros at sv and sν and thus the zeros of the function h are exactly sv and s (symmetrically with respect to the critical line), v = 1, 2, ; ν · · · (c) lim h(s)= 1 ; s→1 2 (d) h satisfies the same equation (1.4) as ξ does, i.e., h(1 s)= h(s). (2.2) − To see (a), it is clear from the definition of ξ in (1.3) that all the zeros of ξ lie in the critical strip and they are zeros of ζ, i.e., ρn. Recall that ξ is of order 1. It follows from Jensen’s formula that n(r, s ) n(r, ρ ) Kr1+ǫ { ν } ≤ { n} ≤ for any ǫ > 0, where K > 0 is a constant and n(r, ρ ) (resp. n(r, s )) { n} { ν } denotes the number of the points ρ ,n = 1, 2, (resp. s ,ν = 1, 2, ) n · · · ν · · · lying in the disc s r (see e.g. [15], p.249). Thus for s 1, | |≤ | |≥ ∞ 2s2 log h(s) log 1+ log 2 | |≤ | s2 | − Xν=1 ν ∞ 2 s2 = log 1+ | 2 | dn(r, sν ) log 2 Z0 r { } − ∞ n(r, s ) = 4 s 2 { ν } dr log 2 | | Z r(r2 + 2 s 2) − 0 | | |s| Kr1+ǫ ∞ Kr1+ǫ 4 s 2 dr + dr log 2 ≤ | | Z 2r s 2 Z r3 − 0 | | |s| 2K 4K = s 1+ǫ + s 1+ǫ log 2, 1+ ǫ| | 1 ǫ| | − − 4 which implies that h is an entire function of order 1. This shows (a). The ≤ 2 2 property (b) is immediate by the fact that (s sv)(s sν) = s s + sν 1 − − − | | since Re(sν) = 2 . The properties (c) and (d) are also immediate, directly from the expression of h in (2.1). Further, we define a meromorphic function η(s) using the same expres- sion as that for ζ in (1.3) (with the role of ξ there being replaced by h), s s s h(s)= (s 1)π− 2 Γ η(s). (2.3) 2 − 2 Replacing s by 1 s yields that − s 1−s 1 s h(1 s)= (s 1)π− 2 Γ − η(1 s), − 2 − 2 − which implies, in view of (2.2) and (2.3), that h(1 s) η(1 s)= −1−s − s − 2 1−s 2 (s 1)π Γ 2 s − s − 2 s 2 (s 1)π Γ 2 η(s) = − 1−s s − 2 1−s 2 (s 1)π Γ 2 −1 −s+ 2 s π Γ( 2 ) = 1−s η(s) Γ( 2 ) πs = 2(2π)−s cos Γ(s)η(s), 2 − 1 s+ 2 s π Γ( 2 ) −s πs by virtue of the identity 1−s = 2(2π) cos 2 Γ(s) (see e.g. [16], Γ( 2 ) p.16). That is, the function η also satisfies the Riemann functional equation (1.2) as ζ does: πs η(1 s) = 2(2π)−s cos Γ(s)η(s).
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