Uniqueness of Dirichlet Series and Algebraic Independence in Selberg Class

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Published on International Journal of Biology, Physics & Matematics
ISSN: 2721-3757, Volume 1, Issue 2, page 1 – 14
Publication Date: 21 November 2018

Ali Mohamed Abuoam Musshim
Faculty of Pure and Applied Sciences, Department of Mathematics and Computer
International University of Africa
Suand or Sudan

Journal Full Text PDF: Uniqueness of Dirichlet Series and Algebraic Independence in Selberg Class.

Abstract
A more precise result is obtained under more restrictive assumptions but still applying to a large class of Dirichlet series. This implies that the equation F^a = G^b with (a, b) = 1 has the unique solution F = H^b and G = H^a in the Selberg class. As a consequence, we show that if F and G are distinct primitive elements of the Selberg class, then the transcendence degree of C[F,G] over C is two.

Keywords: Dirichlet Series, Selberg C, Transcendence Degree, Riemann Zeta Function, L-functions, Zeros and Poles.

1. Introduction
In this paper, we show that how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑_(n=1)^∞▒1/n^s as the prototype and are important objects in number theory. Also, we discuss that under certain mild analytic assumptions one obtains a lower bound, essentially of order r , for the number of zeros and poles of a Dirichlet series in a disk of radius r. A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet Series. More-over we show that the Selberg class S has a natural structure of semi-group.

2. Selberg Class of Dirichlet Series
This section concerns the question of how L-functions are determined by their zeros? L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑_(n=1)^∞▒1/n^s as the prototype and are important objects in number theory. An L_function in the Selberg class S means a Dirichlet series L(s)= ∑_(n=1)^∞▒(a(n))/n^s of a complex variable s = σ +it with a(1) =1, satisfying the following axioms [1]:
(i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series,
(ii) (Analytic continuation) There is a non-negative integer k such that〖(s-1)〗^kL(s) is an entire function of finite order,
(iii) ( Functional equation ) L satisfies a functional equation of type

Λ_L (s)=ω(Λ_L (1-s ̅)) ̅ ,
Where Λ_L (s) = L(s)Q^s ∏_(j=1)^K▒Γ(λ_j s+μ_j) with positive real numbers Q, λ_j , and complex numbers μ_j , ω with Reμ_j ≥ 0 and |ω| = 1,
(iv) (Ramanujan hypothesis) a(n) ≪n^ε for every ε > 0,
(v) (Euler product) logL(s) =∑_(n=1)^∞▒(b(n))/n^s , where b(n) = 0 unless n is a positive power of a prime and b(n) ≪n^θ for some θ <1/2 .
The degree d_(L )of an L-function L is defined to be d_(L ) =2∑_(j=1)^K▒λ_j , where K, λ_j are the numbers in the axiom (iii).
The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis at the same time, there are a whole host of interesting Dirichlet series not possessing Euler product [2], [3]. All L-functions are assumed to be in the extended Selberg class of those only satisfying the axioms (i)-(iii) [3].
Two L-functions with “enough” common zeros are expected to be“dependent” in certain sense (cf. [4]), which, as pointed out in [4], appears to be a very difficult problem. On the other hand, L-functions are meromorphic functions and meromo-rphic functions possess the well-known uniqueness property by Nevanlina’s uniq-ueness theorem two nonconstant meromorphic functions f, g in C must be identic-ally equal if f and g share five distinct values c_j∈ C∪{∞} in the sense that f − c_j and g − c_j have the same zeros without counting multiplicities ( [5] or [2]). For L-functions satisfying the same functional equation and sharing two complex numb-ers c_1, c_2, Steuding proved the uniqueness under a condition on the number of the distinct zeros of L−〖 c〗_j (see [2]). Further considered if this still holds for one shared value and, in particular showed that two L-functions satisfying the same functi-onal equation can be distinct even if they have the same zeros with, of course, diff-erent multiplicities.
An L-function L is completely determined by the functional equation and its nontrivial zeros x, allowing an exceptional set G of x in Z^+(L), which is the set of nontrivial zeros of L counted with multiplicity.As usual nontrivial zeros of an L-function L are those not coming from the poles of the Γ factors in the functional equation. We would like G to be as large as possible. The size of G can be meas-ured by the usual counting function n(r,G), the number of points in G ∩{ |s| < r} counted with multiplicity. It would be tempting to think that the condition n(r,G) = o(r) would be the best to obtain for the exceptional set, since n(r,Z(L)) of an L-function of degree zero is O(r).However, a quite delicate analysis shows that a sharp condition can be given in terms of the type of ……..