# Bicomplex Landau and Ikehara Theorems for the Dirichlet Series

Authors: Ritu Agarwal; Urvashi Purohit Sharma; Ravi P. Agarwal; Daya Lal Suthar; Sunil Dutt Purohit
Journal: Journal of Mathematics
 (2022)

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## Abstract

The aim of this paper is to generalize the Landau-type Tauberian theorem for the bicomplex variables. Our findings extend and improve on previous versions of the Ikehara theorem. Also boundedness result for the bicomplex version of Ikehara–Korevaar theorem is derived. The purpose of this article is to substantially extend the various complex Tauberian theorems for the Dirichlet series to the bicomplex domain.


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## Body

## 1. Introduction

For a long time, bicomplex numbers have been investigated, and a lot of work has been carried out in this area. Bicomplex numbers are introduced by Segre [1] in 1882. Different algebraic and geometric features of bicomplex numbers, as well as their applications, have been the focus of recent research. Many properties and applications of bicomplex numbers have been discovered (see, [2–8]). In recent developments, efforts have been made to extend the integral transforms [9–14], and a number of special functions like [5, 15–19] to the bicomplex variable from their complex counterparts.The aim of this paper is to extend the various complex Tauberian theorems for the Dirichlet series to the bicomplex domain. Generalization of Landau-type theorem and Ikehara theorem is introduced. Boundedness condition for the bicomplex Tauberian theorem has been included. In the proof of these results, the decomposition theorem of Ringleb plays a vital role.


### 1.1. Bicomplex Numbers

The set of bicomplex numbers was defined by Segre [1] in the following way:Definition 1 (Bicomplex number).
The set of bicomplex numbers is defined in terms of real components as(1)T=ξ:ξ=x0+i1x1+i2x2+jx3|x0,x1,x2,x3∈ℝ,and it can be represented as in terms of complex numbers as(2)T=ξ:ξ=z1+i2z2|z1,z2∈ℂ,where i12=i22=−1,i1i2=i2i1=j,j2=1.The notations we will use are as follows:x0=Reξ,x1=Imi1ξ,x2=Imi2ξ,x3=Imjξ.The set of all zero divisor elements ofT is called null cone, and it is denoted by ℕℂ and is defined as follows:(3)ℕℂ=z1+z2i2|z12+z22=0.Segre [1] noticed that the two zero divisor elements 1+i1i2/2 and 1−i1i2/2 are idempotent elements and play a vital role in the theory of the bicomplex numbers. e1 and e2, the two nontrivial idempotent elements of T, are defined as follows:(4)e1=1+i1i22=1+j2,e2=1−i1i22=1−j2.Also,(5)e1+e2=1,e1⋅e2=0,e12=e1,e22=e2.Definition 2 (idempotent representation).
T has a unique idempotent representation for each element [4, 20–22] defined by(6)ξ=z1+i2z2=z1−i1z2e1+z1+i1z2e2.
So, ifξ1=z1−i1z2 and ξ2=z1+i1z2, then ξ=ξ1e1+ξ2e2.Writingξ in real components and idempotent components as(7)ξ=x0+i1x1+i2x2+i1i2x3=x0+i1x1+i2x2+i1x3=ξ1e1+ξ2e2,and comparing them, we get ξ1=x0+x3+i1x1−x2 and ξ2=x0−x3+i1x1+x2.The set of hyperbolic numbersD=x1+x3j|x1,x3∈ℝ,j2=1andj∉ℝ and the set of complex numbers ℂ are two important proper subsets which are unified by the set of bicomplex numbers T (see, [[6], p.19]). The sets T,D are connected to the theory of Clifford algebras. The set of bicomplex number is a two-dimensional complex Clifford algebra which has a set of hyperbolic numbers as its real (Clifford) subalgebra (see [[6], p.24]), or T≅CIℂ1,0≅CIℂ0,1 and D≅CIℝ0,1 (see [[7], p.1]).Definition 3 (bicomplex moduli).
Letξ=z1+i2z2=ξ1e1+ξ2e2=x0+x1i1+x2i2+x3j∈T (see [6, 22, 23]).
The norm ofξ is defined as(8)ξ=z12+z22=12ξ12+ξ22=x02+x12+x22+x32.
Thei1 modulus of ξ is given by(9)ξi1=z12+z22.
Thei2 modulus of ξ is given by(10)ξi2=z12−z22+2Rez1z1¯i2.
Thej modulus of ξ is given by(11)ξj=z1−i1z2e1+z1+i1z2e2.
The absolute value ofξ is given by(12)ξabs=z12+z22=z1−i1z2z1+i1z2=ξ1ξ2=ξ1ξ2.Ringleb [24] (see also [22]), investigated the analyticity of a bicomplex function with respect to its idempotent complex component functions in the following theorem. When studying the convergence of bicomplex functions, this theorem is crucial.Theorem 1 (decomposition theorem of Ringleb [24]).
Letfξ be analytic in a region U⊆T, and let T1⊆ℂ and T2⊆ℂ be the component regions of T, in the ξ1 and ξ2 planes, respectively. Then, there exists a unique pair of complex-valued analytic functions, f1ξ1 and f2ξ2, defined in U1⊆T1 and U2⊆T2, respectively, such that(13)fξ=f1ξ1e1+f2ξ2e2,ξ∈U.Conversely, iff1ξ1 is any complex-valued analytic function in a region T1 and f2ξ2 any complex-valued analytic function in a region T2, then the bicomplex-valued function fξ defined by equation (13) is an analytic function of the bicomplex variable ξ in the product region U=U1×eU2.In 1826, Abel proved the following result for the real power series (see [25–27]).Theorem 2 (Abel’s theorem).
Let(14)fx=∑n=0∞anxn,be a power series with coefficients an∈ℝ that converges on −1,1. We assume that ∑n=0∞an converges. Then,(15)limx⟶1fx=∑n=0∞an.In general, the converse is not true, i.e., iflimx⟶1fx exists, one cannot conclude that ∑n=0∞an converges. In 1897, Tauber [28] proved the converse to Abel’s theorem but under an additional hypothesis.Theorem 3 (Tauberian theorem).
Let(16)fx=∑n=0∞anxn,be a power series with coefficients an∈ℝ that converges on the real interval −1,1. We assume that(17)limx⟶1fx=A,exists, and moreover,(18)limn⟶∞nan=0.Then,∑n=0∞an converges and is equal to A.Detailed proof of the above theorem may be found in [[27], p.435].Tauber’s result directed to many other Tauberian theorems. Later, various other converse theorems have been proved by Hardy and Littlewood and they named them the “Tauberian theorems” (see [26, 29]).Tauberian theory provides many techniques for resolving difficult problems in analysis. Tauberian type theorems have numerous applications in mathematics, including rapidly decaying distributions and their applications to stable laws [30], generalized functions [31], Dirichlet series [32], and the solution of the prime number theorem [26]. In the bicomplex variable [10], the Tauberian theorem for the Laplace–Stieltjes transform is proved. Tauberian theory provides novel answers to complex situations. It has a variety of applications in number theory [26, 33]. In the area of mathematical physics, applications are studied in the quantum field theory [31, 34].Landau [35] (see also [[32], p.4]) studied the following Tauberian result for complex power series.Theorem 4 (Landau’s theorem).
LetG be given for Rew>1,w∈ℂ by a convergent Dirichlet series(19)Gw=∑n=1∞annw,with an≥0, ∀n∈ℕ. We suppose that for some constant α, the analytic function(20)Hw=Gw−αw−1,Rew>1,has an analytic or just continuous extension (also called H) to the closed half-plane Rew≥1. Finally, we suppose that there is a constant K such that(21)Hw=OwK,Rew≥1,K>0.Then,(22)1n∑k=1nak⟶α,asn⟶∞.Ikehara’s theorem [25] extends the result of Landau (see [29]).Theorem 5 (Ikehara’s theorem).
LetG be given by the Dirichlet series(23)Gw=∑n=1∞annw,convergent for Rew>1, where the coefficients satisfy the Tauberian condition an≥0,∀n∈ℕ. If there exists a constant α such that(24)Gw−αw−1,admits a continuous extension to the line Rew=1, then(25)∑k=1nak∽αn,asn⟶∞.In [36, 37], the authors defined the bicomplex Dirichlet series as fξ=∑n=1∞ane−λnξ,ξ∈T, where an, an=α1ne1+α2ne2 is a bicomplex number sequence. Substituting λn=logn, the following form of the bicomplex Dirichlet series is obtained:(26)fξ=∑n=1∞ann−ξ.In terms of idempotent components,fξ can be written as(27)fξ=∑n=1∞ann−ξ=∑n=1∞α1nn−ξ1e1+∑n=1∞α2nn−ξ2e2=f1ξ1e1+f2ξ2e2.The idempotent components offξ, f1ξ1=∑n=1∞α1nn−ξ1 and f2ξ2=∑n=1∞α2nn−ξ2 are the complex Dirichlet Series.If the abscissae of convergence of the seriesf1ξ1=∑n=1∞α1nn−ξ1 and f2ξ2=∑n=1∞α2nn−ξ2 are denoted by σ1 and σ2, respectively, then the region(28)E=ξ∈T:Reξ1>σ1andReξ2>σ2,or equivalently(29)E=ξ∈T:−Reξ+σ1<Imjξ<Reξ−σ2,is the region of convergence of the bicomplex Dirichlet series fξ defined in equation (26).Inspired by the work of Agarwal et al. [10] and Srivastava and Kumar [37], here, the bicomplex Landau-type Tauberian theorem is investigated. Also, the bicomplex version of the Ikehara’s Tauberian theorem, which is generalization of the Landau-type Tauberian theorem, has been studied.


## 1.1. Bicomplex Numbers

The set of bicomplex numbers was defined by Segre [1] in the following way:Definition 1 (Bicomplex number).
The set of bicomplex numbers is defined in terms of real components as(1)T=ξ:ξ=x0+i1x1+i2x2+jx3|x0,x1,x2,x3∈ℝ,and it can be represented as in terms of complex numbers as(2)T=ξ:ξ=z1+i2z2|z1,z2∈ℂ,where i12=i22=−1,i1i2=i2i1=j,j2=1.The notations we will use are as follows:x0=Reξ,x1=Imi1ξ,x2=Imi2ξ,x3=Imjξ.The set of all zero divisor elements ofT is called null cone, and it is denoted by ℕℂ and is defined as follows:(3)ℕℂ=z1+z2i2|z12+z22=0.Segre [1] noticed that the two zero divisor elements 1+i1i2/2 and 1−i1i2/2 are idempotent elements and play a vital role in the theory of the bicomplex numbers. e1 and e2, the two nontrivial idempotent elements of T, are defined as follows:(4)e1=1+i1i22=1+j2,e2=1−i1i22=1−j2.Also,(5)e1+e2=1,e1⋅e2=0,e12=e1,e22=e2.Definition 2 (idempotent representation).
T has a unique idempotent representation for each element [4, 20–22] defined by(6)ξ=z1+i2z2=z1−i1z2e1+z1+i1z2e2.
So, ifξ1=z1−i1z2 and ξ2=z1+i1z2, then ξ=ξ1e1+ξ2e2.Writingξ in real components and idempotent components as(7)ξ=x0+i1x1+i2x2+i1i2x3=x0+i1x1+i2x2+i1x3=ξ1e1+ξ2e2,and comparing them, we get ξ1=x0+x3+i1x1−x2 and ξ2=x0−x3+i1x1+x2.The set of hyperbolic numbersD=x1+x3j|x1,x3∈ℝ,j2=1andj∉ℝ and the set of complex numbers ℂ are two important proper subsets which are unified by the set of bicomplex numbers T (see, [[6], p.19]). The sets T,D are connected to the theory of Clifford algebras. The set of bicomplex number is a two-dimensional complex Clifford algebra which has a set of hyperbolic numbers as its real (Clifford) subalgebra (see [[6], p.24]), or T≅CIℂ1,0≅CIℂ0,1 and D≅CIℝ0,1 (see [[7], p.1]).Definition 3 (bicomplex moduli).
Letξ=z1+i2z2=ξ1e1+ξ2e2=x0+x1i1+x2i2+x3j∈T (see [6, 22, 23]).
The norm ofξ is defined as(8)ξ=z12+z22=12ξ12+ξ22=x02+x12+x22+x32.
Thei1 modulus of ξ is given by(9)ξi1=z12+z22.
Thei2 modulus of ξ is given by(10)ξi2=z12−z22+2Rez1z1¯i2.
Thej modulus of ξ is given by(11)ξj=z1−i1z2e1+z1+i1z2e2.
The absolute value ofξ is given by(12)ξabs=z12+z22=z1−i1z2z1+i1z2=ξ1ξ2=ξ1ξ2.Ringleb [24] (see also [22]), investigated the analyticity of a bicomplex function with respect to its idempotent complex component functions in the following theorem. When studying the convergence of bicomplex functions, this theorem is crucial.Theorem 1 (decomposition theorem of Ringleb [24]).
Letfξ be analytic in a region U⊆T, and let T1⊆ℂ and T2⊆ℂ be the component regions of T, in the ξ1 and ξ2 planes, respectively. Then, there exists a unique pair of complex-valued analytic functions, f1ξ1 and f2ξ2, defined in U1⊆T1 and U2⊆T2, respectively, such that(13)fξ=f1ξ1e1+f2ξ2e2,ξ∈U.Conversely, iff1ξ1 is any complex-valued analytic function in a region T1 and f2ξ2 any complex-valued analytic function in a region T2, then the bicomplex-valued function fξ defined by equation (13) is an analytic function of the bicomplex variable ξ in the product region U=U1×eU2.In 1826, Abel proved the following result for the real power series (see [25–27]).Theorem 2 (Abel’s theorem).
Let(14)fx=∑n=0∞anxn,be a power series with coefficients an∈ℝ that converges on −1,1. We assume that ∑n=0∞an converges. Then,(15)limx⟶1fx=∑n=0∞an.In general, the converse is not true, i.e., iflimx⟶1fx exists, one cannot conclude that ∑n=0∞an converges. In 1897, Tauber [28] proved the converse to Abel’s theorem but under an additional hypothesis.Theorem 3 (Tauberian theorem).
Let(16)fx=∑n=0∞anxn,be a power series with coefficients an∈ℝ that converges on the real interval −1,1. We assume that(17)limx⟶1fx=A,exists, and moreover,(18)limn⟶∞nan=0.Then,∑n=0∞an converges and is equal to A.Detailed proof of the above theorem may be found in [[27], p.435].Tauber’s result directed to many other Tauberian theorems. Later, various other converse theorems have been proved by Hardy and Littlewood and they named them the “Tauberian theorems” (see [26, 29]).Tauberian theory provides many techniques for resolving difficult problems in analysis. Tauberian type theorems have numerous applications in mathematics, including rapidly decaying distributions and their applications to stable laws [30], generalized functions [31], Dirichlet series [32], and the solution of the prime number theorem [26]. In the bicomplex variable [10], the Tauberian theorem for the Laplace–Stieltjes transform is proved. Tauberian theory provides novel answers to complex situations. It has a variety of applications in number theory [26, 33]. In the area of mathematical physics, applications are studied in the quantum field theory [31, 34].Landau [35] (see also [[32], p.4]) studied the following Tauberian result for complex power series.Theorem 4 (Landau’s theorem).
LetG be given for Rew>1,w∈ℂ by a convergent Dirichlet series(19)Gw=∑n=1∞annw,with an≥0, ∀n∈ℕ. We suppose that for some constant α, the analytic function(20)Hw=Gw−αw−1,Rew>1,has an analytic or just continuous extension (also called H) to the closed half-plane Rew≥1. Finally, we suppose that there is a constant K such that(21)Hw=OwK,Rew≥1,K>0.Then,(22)1n∑k=1nak⟶α,asn⟶∞.Ikehara’s theorem [25] extends the result of Landau (see [29]).Theorem 5 (Ikehara’s theorem).
LetG be given by the Dirichlet series(23)Gw=∑n=1∞annw,convergent for Rew>1, where the coefficients satisfy the Tauberian condition an≥0,∀n∈ℕ. If there exists a constant α such that(24)Gw−αw−1,admits a continuous extension to the line Rew=1, then(25)∑k=1nak∽αn,asn⟶∞.In [36, 37], the authors defined the bicomplex Dirichlet series as fξ=∑n=1∞ane−λnξ,ξ∈T, where an, an=α1ne1+α2ne2 is a bicomplex number sequence. Substituting λn=logn, the following form of the bicomplex Dirichlet series is obtained:(26)fξ=∑n=1∞ann−ξ.In terms of idempotent components,fξ can be written as(27)fξ=∑n=1∞ann−ξ=∑n=1∞α1nn−ξ1e1+∑n=1∞α2nn−ξ2e2=f1ξ1e1+f2ξ2e2.The idempotent components offξ, f1ξ1=∑n=1∞α1nn−ξ1 and f2ξ2=∑n=1∞α2nn−ξ2 are the complex Dirichlet Series.If the abscissae of convergence of the seriesf1ξ1=∑n=1∞α1nn−ξ1 and f2ξ2=∑n=1∞α2nn−ξ2 are denoted by σ1 and σ2, respectively, then the region(28)E=ξ∈T:Reξ1>σ1andReξ2>σ2,or equivalently(29)E=ξ∈T:−Reξ+σ1<Imjξ<Reξ−σ2,is the region of convergence of the bicomplex Dirichlet series fξ defined in equation (26).Inspired by the work of Agarwal et al. [10] and Srivastava and Kumar [37], here, the bicomplex Landau-type Tauberian theorem is investigated. Also, the bicomplex version of the Ikehara’s Tauberian theorem, which is generalization of the Landau-type Tauberian theorem, has been studied.


## 2. Bicomplex Versions of the Landau and Ikehara Theorems

Motivated by the work of Landau, we have derived the bicomplex version of Theorem4 as follows:Theorem 6 (bicomplex Landau theorem).
Letf be given for ξ=ξ1e1+ξ2e2, Imjξ<Reξ−1 by a convergent Dirichlet series(30)fξ=∑n=1∞annξ,ξ,an∈T,where an=an1+jan4∈D with an1≥an4, ∀n∈ℕ. We suppose that for some hyperbolic constant A=A1e1+A2e2, the analytic function(31)gξ=fξ−Aξ−1,Imjξ<Reξ−1,has an analytic or just continuous extension (also called g) to the closed half-plane Imjξ≤Reξ−1.Finally, we suppose that there is a constantM such that(32)gξ=OξjM,for Imjξ≤Reξ−1. Then,(33)1nSn=1n∑k=1nak⟶A,asn⟶∞.Proof.
We consider the Dirichlet series(34)fξ=∑n=1∞annξ=fξ1e1+fξ2e2,an,ξ∈T,and let A1e1+A2e2=A∈T. Here,(35)fξ1=∑n=1∞α1nnξ1,Reα1n>0,fξ2=∑n=1∞α2nnξ2,Reα2n>0,are convergent for Reξ1>1 and Reξ2>1, respectively.
For some constantsAi,i=1,2,(36)giξi=fξi−Aiξi−1,Reξi>1,i=1,2,are analytic functions in the complex domain. By Theorem 4, function giξi,i=1,2 has an analytic or just continuous extension (also called gi) to the closed half plane Re ξi≥1,i=1,2.
Sinceg1ξ1 and g2ξ2 are analytic functions, thereby taking the idempotent linear combination of (36) for i=1,2,(37)gξ=g1ξ1e1+g2ξ2e2=fξ1−A1ξ1−1e1+fξ2−A2ξ2−1e2=fξ1e1+fξ2e2−A1e1+A2e2ξ1e1+ξ2e2−1=fξ−Aξ−1.
With the help of equation (7), the conditions Reξ1>1,Reξ2>1 can be rewritten as(38)x0+x3>1,x0−x3>1,⇒x3<x0−1⇒Imjξ<Reξ−1.
By assumption of the theorem, the j-modulus ofgξ, in (37), ξ∈T, we have(39)gξ=OξjM,Since,ξjM=ξ1Me1+ξ2Me2,⇒gξ1=Oξ1Mgξ2=Oξ2M.
Thus, by Theorem4 for complex domain,(40)1nS1n=1n∑k=1nα1k⟶A1,1nS2n=1n∑k=1nα2k⟶A2,asn⟶∞.
By idempotent combination of the above series,(41)1nS1ne1+1nS2ne2=1nSn⟶A1e1+A2e2=A,1nSn⟶A,asn⟶∞.
Furthermore, the relationan=α1ne1+α2ne2=an1+i1an2+i2an3+jan4 gives(42)α1n=an1+an4+i1an2−an3,α2n=an1−an4+i1an2+an3.
The conditionsα1n≥0, α2n≥0 imply(43)an1+an4≥0,an2−an3=0;an1−an4≥0,an2+an3=0.⇒an1≥an4;an2=an3=0.⇒an=an1+an4e1+an1−an4e2=an1+jan4.
Hence,an is a hyperbolic number with an1≥an4.Remark 1.
In the proof of the above theorem, it is observed that the results and conditions focus on the hyperbolic coefficients and not on coefficients of imaginary unitsi1 and i2; hence, it can be called the hyperbolic version of the Landau theorem.Theorem 7 (bicomplex Ikehara theorem).
Letξ,an∈T where ξ=ξ1e1+ξ2e2 and an=an1+jan4,n∈ℕ is a sequence of hyperbolic numbers [6]. Let f be given by the Dirichlet series(44)fξ=∑n=1∞annξ,an1≥an4,convergent for Imjξ<Reξ−1.If there exists a hyperbolic constantβ∈D such that(45)fξ−βξ−1,admits a continuous extension to the plane Reξ=1,Imjξ=0, then(46)Sn=∑k=1nan∽βn,asn⟶∞.Proof.
We consider the Dirichlet Series(47)fξ=∑n=1∞annξ=fξ1e1+fξ2e2,where(48)fξi=∑n=1∞αinnξi,αin≥0,Reξi>1,i=1,2,where fξ is convergent for α1n≥0,α2n≥0i.e.an1≥an4 (from equation (43)) and Imjξ<Reξ−1.
By Theorem5, for some constants βi,i=1,2, the analytic functions(49)fiξi−βiξi−1,i=1,2,admit a continuous extension to the lines Re ξi=1,i=1,2. Taking idempotent linear combination of the functions defined in equation (49), we get for Reξi>1,i=1,2 or equivalently Imjξ<Reξ−1,(50)fξ1−β1ξ1−1e1+fξ2−β2ξ2−1e2=fξ−βξ−1,where β=β1e1+β2e2∈D. Hence, fξ−β/ξ−1 admits a continuous extension to the plane Reξ1=1,Reξ2=1,i.e.,Reξ=1,Imjξ=0 which means ξ=1+x1i1+x2i2,x1,x2∈ℝ.
Furthermore,(51)∑k=1nak=∑k=1nα1ke1+∑n=1nα2ke2∽βn,asn⟶∞.


## 3. Ikehara’s Theorem Involving Boundedness

In this section, we discuss some results about Schwartz functions, tempered distributions, and the Fourier transform (see [38–40]). Schwartz [41] (see also [38]) chooses the class of test function ϕ that is infinitely continuously differentiable and that vanishes outside some bounded set. All functionals defined on this class that are linear and continuous are named distributions by Schwartz.SpaceSℝ is the Schwartz space of rapidly decreasing smooth test functions ϕ (see [29]), i.e., those C∞ functions over the real field such that(52)supu∈Rupϕqu<∞,p,q∈ℕ.The space of tempered distributions is represented byS′ℝ, which is the dual of Sℝ (see [29]). The evaluation of g∈S′ℝ at ψ∈Sℝ is denoted by g,ψ=∫−∞∞guψudu. Thus, g∈S′ℝ if and only if(53)g,aψ+ϕ=ag,ψ+g,ϕ,limn⟶∞g,ψn=g,limn⟶∞ψn,whenever ψnn=0∞ is convergent in Sℝ.If a tempered distribution is the Fourier transform of a bounded (measurable) function, then it is called a pseudomeasure.Let∑n=1∞an/nw be a complex Dirichlet series with coefficients an≥0 that converges to a function fw for Rew>1. In 2008, Korevaar [42] proved following theorem for boundedness of SN/N in complex space as follows:Theorem 8 (Ikehara–Korevaar theorem).
Let∑n=1∞an/nw be a Dirichlet series with coefficients an≥0 converging to gw for Rew>1. Let SN=∑n≤Nan; the sequence SN/N will remain bounded if the quotient(54)fw=gww,w=u+i1v∈ℂ,Rew=u>1,converges in the sense of tempered distribution to a pseudomeasure f1+i1v, as u⟶1.Remark 2.
The distributional convergence in the above theorem is convergence in the Schwartz spaceS′. In other words,(55)<fu+i1v,ϕv⟶f1+i1v,ϕv>,asu⟶1,for all testing functions ϕv∈S, that is, all rapidly decreasing C∞ functions.We hereby provide the bicomplex version of Theorem8.Theorem 9 (bicomplex Ikehara–Korevaar theorem).
Let∑n=1∞an/nξ be a bicomplex Dirichlet series where ξ=ξ1e1+ξ2e2=z1−i1z2e1+z1+i1z2e2∈T, z1=x1+i1y1,z2=x2+i1y2∈ℂ, and an=an1+jan4∈D with an1≥an4 that converges to fξ for Imjξ<Reξ−1.LetSN=∑n≤Nan; then, a necessary and sufficient condition for the boundedness of SN/N is that the quotient qξ=fξ/ξ,ξ∉ℕℂ converges in the sense of tempered distribution to a pseudomeasure q1+i1y1+i2x2, as x1⟶1,y2⟶0.Proof.
Let the Dirichlet series∑n=1∞an/nξ converges to fξ=f1ξ1e1+f2ξ2e2 where(56)∑n=1∞annξ=∑n=1∞α1nnξ1e1+∑n=1∞α2nnξ2e2,an=α1ne1+α2ne2.
ForReξ1>1 and Reξ2>1, equivalently, Imjξ<Reξ−1, the Dirichlet series ∑n=1∞α1n/nξ1,α1n≥0 converges to the function f1ξ1 and the Dirichlet series ∑n=1∞α2n/nξ2,α2n≥0 converges to the function f2ξ2.
Let us denote∑n≤Nα1n=S1N and ∑n≤Nα2n=S2N; then,(57)SN=∑n≤Nan=∑n≤Nα1ne1+∑n≤Nα2ne2=S1Ne1+S2Ne2.
From Theorem8, the necessary and sufficient condition for the boundedness of S1N/N is that the quotient(58)q1z1−i1z2=q1x1+y2+i1y1−x2=f1z1−i1z2z1−i1z2,converges in the sense of tempered distribution to a pseudomeasure q11+i1y1−x2, as x1+y2⟶1.
Similarly, the necessary and sufficient condition for boundedness ofS2N/N is that the quotient(59)q2z1+i1z2=q2x1−y2+i1y1+x2=f2z1+i1z2z1+i1z2,converges in the sense of tempered distribution to a pseudomeasure q21+i1y1+x2, as x1−y2⟶1.
Again, by the application of the Ringleb theorem, the necessary and sufficient condition for the boundedness ofSN/N=S1N/Ne1+S2N/Ne2 is that the quotient(60)qz1+i2z2=q1e1+q2e2=f1z1−i1z2z1−i1z2e1+f2z1+i1z2z1+i1z2e2=f1z1−i1z2e1+f2z1+i1z2e21z1+i2z2=fξξ,ξ∉ℕℂ,converges to q11+i1y1−x2e1+q21+i1y1+x2e2=q1+i1y1+i2x2 in the sense of tempered distribution to a pseudomeasure as x1+y2⟶1 and x1−y2⟶1,i.e.,x1⟶1,y2⟶0.


## 4. Conclusion

In this paper, Landau-type Tauberian theorem in bicomplex space which is the generalization of Landau-type Tauberian theorem has been derived. The necessary and sufficient condition for the boundedness of the partial sumSN=∑n≤Nan for bicomplex Dirichlet series with hyperbolic coefficients is obtained. The conditions of convergence are affected by the j coefficient of bicomplex numbers, and hence the theorems can be seen as the hyperbolic versions.



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