Uniqueness of Dirichlet Series and Algebraic Independence in Selberg Class

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

Ali Mohamed Abuoam Musshim

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

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 G

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