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J. Ana. Num. Theor. 5, No. 1, 41-48 (2017) 41 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.18576/jant/050107 Dirichlet-Power Series Hunduma Legesse Geleta∗ Department of Mathematics, Addis Ababa University, Ethiopia Received: 2 Sep. 2016, Revised: 18 Nov. 2016, Accepted: 23 Nov. 2016 Published online: 1 Jan. 2017 Abstract: In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power” series. This series includes both Dirichlet series and power series as a special case. It arose during the investigation of series representations of the second order hypergeometric zeta function. The Dirichlet-power series has many properties analogous to Dirichlet series. For a Dirichlet-power series we establish Domain of convergence and show the existence of zero free regions to the right half plane. Keywords: Multiplicative cone b-metric space, fixed point, contractive mapping. 1 Introduction In this paper we introduce and investigate a new type of series which we call a ”Dirichlet-power series”. This The Riemann Zeta function ζ(s) defined by series includes both Dirichlet series and power series as a ∞ special case. As in the case of Dirichlet series it has a 1 kind of abscissa of absolute convergence is attached to it ∑ s n=1 n with a right half plane convergence. On the other hand as a power series it has a kind of radius of convergence with with an integral representation D-shaped region of convergence. This kind of series actually arose during the investigation of series ∞ s−1 ζ 1 x representations of the second order hypergeometric zeta (s)= x dx Γ (s) Z0 e − 1 function. The hypergeometric zeta function of order N where N is a positive integer, defined as is a special example of a type of series called Dirichlet series. A Dirichlet series is a series of the form ∞ s+N−2 ζ 1 x N (s)= x dx ∞ Γ (s + N − 1) Z0 e − TN−1(x) an ∑ s n=1 n which admits a Dirichlet-type series given by, with a complex number and s complex variable. ∞ n µN (n,s) For a Dirichlet series of this type there is an extended real ζN (s)= ∑ ns+N−1 number σa in the extendedreal line with the propertythat n=1 the series convergesabsolutely for Re(s) > σ , butnotfor a for σ > 1, where Re(s) < σa. Moreover, for any positive number ε, the convergence is uniform for Re(s) > σ + ε, so that the a (N−1)(n−1) a (N,n)Γ (s + N + k − 1) series represents a holomorphic function for all Re(s) > µ (n,s)= ∑ k N kΓ σa. The quantity σa is called the abscissa of absolute k=0 n (s + N − 1) convergence of the Dirichlet series; it is an analogue of the radius of convergence of a power series [8]. In fact, and ak(N,n) is generated by n−1 n−1 x2 x3 xN−1 ∑(N−1)(n−1) k if we fix a prime p, and only allow an to be nonzero (TN−1(x)) = 1 + x + 2! + 3! + ··· + (N−1)! = k=0 ak(N,n)x when n is a power of p, then we get an ordinary power series in p−s. So in some sense, Dirichlet series are a Originally introduced and investigated by Abdul strict generalization of an ordinary power series. Hassen and Heiu D. Nuygen as a generalization of the ∗ Corresponding author e-mail: [email protected] c 2017 NSP Natural Sciences Publishing Cor. 42 H. L. Geleta: Dirichlet-Power Series ∞ Riemann Zeta function, via the integral representation in Suppose there exists an infinite sequence {sk}k=1 of several papers like [2],[4], [5]. Recently we studied complex numbers, with σk = Re(sk) > σ0 for series representation of the second order hypergeometric all k and σk → ∞ such that D(sk)= E(sk) for all k . zeta function in [6] which can be given as: Then D(s)= E(s) for all s with σ = Re(s) > σ0 . ∞ Corollary 23 Let ζ m−1 2(s)= ∑ Dm(s)s ∞ n=1 an D(s)= ∑ s n=1 n where the coefficient Dm(s) is a Dirichlet series for each m = 1,2,3,···. This series representation of the second be a Dirichlet series with abscissa of absolute order hypergeometric zeta function is a source for the title convergence σa . Suppose that for some s with of this paper. σ = Re(s) > σa wehave D(s) 6= 0 . Then there exists a The Dirichlet-power series has many properties half-plane in which the Diriclet series is absolutely analogous to Dirichlet series. For a Dirichlet-power series convergent and never zero. we have a D-shaped region of absolute convergence which can be extended to right half plane. In this From the these properties of Dirichlet series it is known that the Domain of absolute convergence of a extension there is a unique extended real number σa in the extended real line with the property that the series Dirichlet series is one of the following . converges absolutely for Re(s) > σa, but not for 1.empty set, that is for no s does the series absolutely Re(s) < σa just analogous to Dirichlet series. In any converges. compact subset of its absolute convergence the series 2.(−∞,∞), that is the series convergesfor every complex represents an analytic function. Moreover; this series has number s. a zero free regions in the right half-plane. 3.(σ,∞) that is the series converges for every complex number s with Re(s) > σ 4.[σ,∞) that is the series converges for every complex 2 Preliminaries and Auxiliary Results number s with Re(s) ≥ σ Notice that in all cases, there is a unique number σa ∈ In this section we review some main known results on [−∞,∞] such that for all s with Re(s) > σa , the Dirichlet Dirichlet series which we need for our consumption, series is absolutely convergent. And for all s with Re(s) < formally define a Dirichlet-power series and discuss on σa , the Dirichlet series is not absolutely convergent. the domain of absolute convergence. 2.2 Definition and Examples of Dirichlet-Power 2.1 Review on Dirichlet Series Series We summarize some of the properties of Dirichlet series Definition 24 Let Dm(s) be a Dirichlet series for each m = as follows: 1,2,.... We define a Dirichlet-Power series to be a series of the form Theorem 21 Suppose a Dirichlet series ∞ m−1 ∞ ∑ Dm(s)s (2.1) −s m=1 D(s)= ∑ amm m=1 Remark 25 is absolutely convergent at a point s0 = σ0 + it0. Then it is also absolutely convergent at all points s with σ = 1.A Dirichlet-power series has a form of a power series Re(s) > σ . but with a Dirichlet series as a coefficient. 0 2.If a Dirichlet-power series converges absolutely Theorem 22 (Uniqueness theorem for Dirichlet series) for Re(s) > σ0, then Dm(s) converges absolutely for each m = 1,2,3,..., for Re(s) > σ0. Let ∞ 3.If Dm(s) is identically zero for m ≥ 2, then the an Dirichlet-power series becomes a Dirichlet series. D(s)= ∑ s 4.If D s D a constant sequence, then the n=1 n m( ) = m, Dirichlet-power series becomes a power series. and ∞ b Proposition 26 Suppose the function F(s) is represented E(s)= ∑ n s by a Dirichlet-power series, for Re(s) > σa, and suppose n=1 n ∞ where D(s) and E(s) are both absolutely convergent m−1 σ σ ∑ Dm(s)s → 0 Dirichlet series in the half-plane = Re(s) > 0 . m=2 c 2017 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 5, No. 1, 41-48 (2017) / www.naturalspublishing.com/Journals.asp 43 as σ = Re(s) → ∞, independent of t. Then Theorem 28 Suppose the series F(s) → a ∞ 11 m−1 ∑ |Dm(s)s | as σ = Re(s) → ∞ m=1 Proof. By the hypothesis of the proposition we have, does not converge for all t or diverge for all t on a line ∞ with Re(s)= σ0. Then there exists a real number t0, such m−1 ∑ Dm(s)s → 0 that the series ∞ m=2 m−1 ∑ |Dm(s)s | as σ = Re(s) → ∞, independent of t. Then we are left only m=1 with the first term of the Dirichlet-power series D1. But D1 converges for all t such that 0 ≤ t ≤ t0 and for any s with is a Dirichlet series and hence by the property of Dirichlet σ σ σ 2 2 σ 2 2 > 0 with the property +t < a +ta , but does not series it converges to its first term which is a11 independent converge otherwise. of t as σ = Re(s) → ∞. Proof. Let E = Some examples of a Dirichlet-power series are given here: ∑∞ m−1 {t/t is nonnegative and m=1 |Dm(s)s |converges} 1.the second order Hypergeometric zeta function given where s = σ0 + it. Let t0 be the supremum of t in E such by, that ∞ ∞ m−1 m−1 ζ2(s)= ∑ Dm(s)s ∑ |Dm(s)s |converges} n=1 m=1 where the coefficient Dm(s) is a Dirichlet series for Such supremum exists since the series does not converge each m = 1,2,3,··· , where the Dm(s) is given as in [6]. for all t. 2.a series given by, Observe that from the theorem we have a segment E = ζ s F(s)= (s)e {σ0 + it/ − t0 ≤ t ≤ t0}.