Bicomplex Landau and Ikehara Theorems for the Dirichlet Series

Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India Department of Science and Humanities, Kalaniketan Polytechnic College, Jabalpur-482001, India Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, USA Department of Mathematics, Wollo University, P.O. Box: 1145, Dessie, Ethiopia Department of HEAS(Mathematics), Rajasthan Technical University, Kota-324010, India


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][3][4][5][6][7][8]). In recent developments, efforts have been made to extend the integral transforms [9][10][11][12][13][14], and a number of special functions like [5,[15][16][17][18][19] to the bicomplex variable from their complex counterparts. e 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.

Bicomplex Numbers.
e set of bicomplex numbers was defined by Segre [1] in the following way: Definition 1 (Bicomplex number). e set of bicomplex numbers is defined in terms of real components as (1) and it can be represented as in terms of complex numbers as where i 2 1 � i 2 2 � − 1, i 1 i 2 � i 2 i 1 � j, j 2 � 1.
e notations we will use are as follows: e set of all zero divisor elements of T is called null cone, and it is denoted by NC and is defined as follows: Segre [1] noticed that the two zero divisor elements (1 + i 1 i 2 )/2 and (1 − i 1 i 2 )/2 are idempotent elements and play a vital role in the theory of the bicomplex numbers. e 1 and e 2 , the two nontrivial idempotent elements of T, are defined as follows: Also, Definition 2 (idempotent representation). T has a unique idempotent representation for each element [4,[20][21][22] defined by Writing ξ in real components and idempotent components as and comparing them, we get . e set of hyperbolic numbers D � x 1 + x 3 j|x 1 , x 3 ∈ R, j 2 � 1 and j ∉ 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]). e sets T, D are connected to the theory of Clifford algebras. e 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 � CI C (1, 0) � CI C (0, 1) and D � CI R (0, 1) (see [[7], p.1]). [6,22,23]). e norm of ξ is defined as e i 1 modulus of ξ is given by e i 2 modulus of ξ is given by e j modulus of ξ is given by e absolute value of ξ is given by 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]). Let f(ξ) be analytic in a region U⊆T, and let T 1 ⊆C and T 2 ⊆C be the component regions of T, in the ξ 1 and ξ 2 planes, respectively. en, there exists a unique pair of complex-valued analytic functions, f 1 (ξ 1 ) and f 2 (ξ 2 ), defined in U 1 ⊆T 1 and U 2 ⊆T 2 , respectively, such that Conversely, if f 1 (ξ 1 ) is any complex-valued analytic function in a region T 1 and f 2 (ξ 2 ) any complex-valued analytic function in a region T 2 , then the bicomplex-valued function f(ξ) defined by equation (13) is an analytic function of the bicomplex variable ξ in the product region U � U 1 × e U 2 .

Theorem 2 (Abel's theorem). Let
be a power series with coefficients a n ∈ R that converges on (− 1, 1). We assume that ∞ n�0 a n converges. en, In general, the converse is not true, i.e., if lim x⟶1 f(x) exists, one cannot conclude that ∞ n�0 a n converges. In 1897, Tauber [28] proved the converse to Abel's theorem but under an additional hypothesis.

Theorem 3 (Tauberian theorem). Let
be a power series with coefficients a n ∈ R that converges on the real interval (− 1, 1). We assume that exists, and moreover, en, ∞ n�0 a n 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].
Theorem 4 (Landau's theorem). Let G be given for Re(w) > 1, w ∈ C by a convergent Dirichlet series with a n ≥ 0, ∀n ∈ N. We suppose that for some constant α, the analytic function has an analytic or just continuous extension (also called H) to the closed half-plane Re(w) ≥ 1. Finally, we suppose that there is a constant K such that en, Ikehara's theorem [25] extends the result of Landau (see [29]).

Theorem 5 (Ikehara's theorem). Let G be given by the Dirichlet series
convergent for Re(w) > 1, where the coefficients satisfy the Tauberian condition a n ≥ 0, ∀n ∈ N. If there exists a constant α such that admits a continuous extension to the line Re(w) � 1, then n k�1 a k ∽αn, as n ⟶ ∞.

Bicomplex Versions of the Landau and Ikehara Theorems
Motivated by the work of Landau, we have derived the bicomplex version of eorem 4 as follows: Theorem 6 (bicomplex Landau theorem). Let f be given for ξ � ξ 1 e 1 + ξ 2 e 2 , |Im j (ξ)| < Re(ξ) − 1 by a convergent Dirichlet series f(ξ) � ∞ n�1 a n n ξ , ξ, a n ∈ T, where a n � a n 1 + ja n 4 ∈ D with a n 1 ≥ |a n 4 |, ∀n ∈ N. We suppose that for some hyperbolic constant A � A 1 e 1 + A 2 e 2 , the analytic function has an analytic or just continuous extension (also called g) to the closed half-plane |Im j (ξ)| ≤ Re(ξ) − 1.
Finally, we suppose that there is a constant M such that for |Im j (ξ)| ≤ Re(ξ) − 1. en, 1 n S n � 1 n n k�1 a k ⟶ A, as n ⟶ ∞.
⇒a n � a n 1 + a n 4 e 1 + a n 1 − a n 4 e 2 � a n 1 + ja n 4 .

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Hence, a n is a hyperbolic number with a n 1 ≥ |a n 4 |. □ 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 units i 1 and i 2 ; hence, it can be called the hyperbolic version of the Landau theorem.
Theorem 7 (bicomplex Ikehara theorem). Let ξ, a n ∈ T where ξ � ξ 1 e 1 + ξ 2 e 2 and a n � a n 1 + ja n 4 , n ∈ N is a sequence of hyperbolic numbers [6]. Let f be given by the Dirichlet series f(ξ) � ∞ n�1 a n n ξ , a n 1 ≥ a n 4 , If there exists a hyperbolic constant β ∈ D such that admits a continuous extension to the plane Re(ξ) � 1, Im j (ξ) � 0, then S n � n k�1 a n ∽βn, as n ⟶ ∞.

Ikehara's Theorem Involving Boundedness
In this section, we discuss some results about Schwartz functions, tempered distributions, and the Fourier transform (see [38][39][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. Space S(R) is the Schwartz space of rapidly decreasing smooth test functions ϕ (see [29]), i.e., those C ∞ functions over the real field such that sup u∈R u p ϕ (q) (u) < ∞, p, q ∈ N.
(52) e space of tempered distributions is represented by S ′ (R), which is the dual of S(R) (see [29]). e evaluation of whenever ψ n ∞ n�0 is convergent in S(R). If a tempered distribution is the Fourier transform of a bounded (measurable) function, then it is called a pseudomeasure.
Let ∞ n�1 a n /n w be a complex Dirichlet series with coefficients a n ≥ 0 that converges to a function f(w) for Re(w) > 1. In 2008, Korevaar [42] proved following theorem for boundedness of S N /N in complex space as follows: Theorem 8 (Ikehara-Korevaar theorem). Let ∞ n�1 a n /n w be a Dirichlet series with coefficients a n ≥ 0 converging to g(w) for Re(w) > 1. Let S N � n≤N a n ; the sequence S N /N will remain bounded if the quotient converges in the sense of tempered distribution to a pseu- Remark 2. e distributional convergence in the above theorem is convergence in the Schwartz space S ′ . In other words, for all testing functions ϕ(v) ∈ S, that is, all rapidly decreasing C ∞ functions.
We hereby provide the bicomplex version of eorem 8.
Let us denote n≤N α 1n � S 1N and n≤N α 2n � S 2N ; then, From eorem 8, the necessary and sufficient condition for the boundedness of S 1N /N is that the quotient converges in the sense of tempered distribution to a pseudomeasure q 1 (1 + i 1 (y 1 − x 2 )), as x 1 + y 2 ⟶ 1. Similarly, the necessary and sufficient condition for boundedness of S 2N /N is that the quotient converges in the sense of tempered distribution to a pseudomeasure q 2 (1 + i 1 (y 1 + x 2 )), as x 1 − y 2 ⟶ 1. Again, by the application of the Ringleb theorem, the necessary and sufficient condition for the boundedness of S N /N � (S 1N /N)e 1 + (S 2N /N)e 2 is that the quotient converges to q 1 (1 + i 1 (y 1 − x 2 ))e 1 + q 2 (1 + i 1 (y 1 + x 2 ))e 2 � q(1 + i 1 y 1 + i 2 x 2 ) in the sense of tempered distribution to a pseudomeasure as x 1 + y 2 ⟶ 1 and x 1 − y 2 ⟶ 1, i.e., x 1 ⟶ 1, y 2 ⟶ 0.

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

Data Availability
No data were used to support this study.