Bicomplex Landau and Ikehara Theorems for the Dirichlet Series

Agarwal, Ritu; Sharma, Urvashi Purohit; Agarwal, Ravi P.; Suthar, Daya Lal; Purohit, Sunil Dutt

doi:https://doi.org/10.1155/2022/4528209

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Advances in Geometric Function Theory

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Volume 2022 | Article ID 4528209 | https://doi.org/10.1155/2022/4528209

Bicomplex Landau and Ikehara Theorems for the Dirichlet Series

Ritu Agarwal,¹Urvashi Purohit Sharma,^1,2Ravi P. Agarwal,³Daya Lal Suthar,⁴and Sunil Dutt Purohit⁵

Academic Editor: V. Ravichandran

Received06 Jan 2022

Revised11 Feb 2022

Accepted14 Feb 2022

Published15 Mar 2022

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.

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 asand it can be represented as in terms of complex numbers aswhere

The notations we will use are as follows:

.

The set of all zero divisor elements of is called null cone, and it is denoted by and is defined as follows:

Segre [1] noticed that the two zero divisor elements and are idempotent elements and play a vital role in the theory of the bicomplex numbers. and , the two nontrivial idempotent elements of , are defined as follows:

Also,

Definition 2 (idempotent representation). has a unique idempotent representation for each element [4, 20–22] defined bySo, if and , then

Writing in real components and idempotent components asand comparing them, we get and .

The set of hyperbolic numbers and the set of complex numbers are two important proper subsets which are unified by the set of bicomplex numbers (see, [[6], p.19]). The sets 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 and (see [[7], p.1]).

Definition 3 (bicomplex moduli). Let (see [6, 22, 23]).
The norm of is defined asThe modulus of is given byThe modulus of is given byThe modulus of is given byThe absolute value of 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 be analytic in a region , and let and be the component regions of , in the and planes, respectively. Then, there exists a unique pair of complex-valued analytic functions, and , defined in and , respectively, such that

Conversely, if is any complex-valued analytic function in a region and any complex-valued analytic function in a region , then the bicomplex-valued function defined by equation (13) is an analytic function of the bicomplex variable in the product region .

In 1826, Abel proved the following result for the real power series (see [25–27]).

Theorem 2 (Abel’s theorem). Letbe a power series with coefficients that converges on . We assume that converges. Then,

In general, the converse is not true, i.e., if exists, one cannot conclude that converges. In 1897, Tauber [28] proved the converse to Abel’s theorem but under an additional hypothesis.

Theorem 3 (Tauberian theorem). Letbe a power series with coefficients that converges on the real interval . We assume thatexists, and moreover,

Then, converges and is equal to .

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). Let be given for by a convergent Dirichlet serieswith , . We suppose that for some constant , the analytic functionhas an analytic or just continuous extension (also called ) to the closed half-plane . Finally, we suppose that there is a constant such that

Then,

Ikehara’s theorem [25] extends the result of Landau (see [29]).

Theorem 5 (Ikehara’s theorem). Let be given by the Dirichlet seriesconvergent for , where the coefficients satisfy the Tauberian condition . If there exists a constant such thatadmits a continuous extension to the line , then

In [36, 37], the authors defined the bicomplex Dirichlet series as where , is a bicomplex number sequence. Substituting the following form of the bicomplex Dirichlet series is obtained:

In terms of idempotent components, can be written as

The idempotent components of , and are the complex Dirichlet Series.

If the abscissae of convergence of the series and are denoted by and , respectively, then the regionor equivalentlyis the region of convergence of the bicomplex Dirichlet series 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 Theorem 4 as follows:

Theorem 6 (bicomplex Landau theorem). Let be given for , by a convergent Dirichlet serieswhere with , . We suppose that for some hyperbolic constant , the analytic functionhas an analytic or just continuous extension (also called ) to the closed half-plane .

Finally, we suppose that there is a constant such thatfor . Then,

Proof. We consider the Dirichlet seriesand let . Here,are convergent for and , respectively.
For some constants ,are analytic functions in the complex domain. By Theorem 4, function has an analytic or just continuous extension (also called ) to the closed half plane Re .
Since and are analytic functions, thereby taking the idempotent linear combination of (36) for ,With the help of equation (7), the conditions can be rewritten asBy assumption of the theorem, the j-modulus of , in (37), , we haveThus, by Theorem 4 for complex domain,By idempotent combination of the above series,Furthermore, the relation givesThe conditions , implyHence, is a hyperbolic number with .

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 and ; hence, it can be called the hyperbolic version of the Landau theorem.

Theorem 7 (bicomplex Ikehara theorem). Let where and is a sequence of hyperbolic numbers [6]. Let be given by the Dirichlet seriesconvergent for .

If there exists a hyperbolic constant such thatadmits a continuous extension to the plane , then

Proof. We consider the Dirichlet Serieswherewhere is convergent for (from equation (43)) and .
By Theorem 5, for some constants the analytic functionsadmit a continuous extension to the lines Re . Taking idempotent linear combination of the functions defined in equation (49), we get for or equivalently ,where . Hence, admits a continuous extension to the plane which means .
Furthermore,

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 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 is the Schwartz space of rapidly decreasing smooth test functions (see [29]), i.e., those functions over the real field such that

The space of tempered distributions is represented by , which is the dual of (see [29]). The evaluation of at is denoted by . Thus, if and only ifwhenever is convergent in .

If a tempered distribution is the Fourier transform of a bounded (measurable) function, then it is called a pseudomeasure.

Let be a complex Dirichlet series with coefficients that converges to a function for . In 2008, Korevaar [42] proved following theorem for boundedness of in complex space as follows:

Theorem 8 (Ikehara–Korevaar theorem). Let be a Dirichlet series with coefficients converging to for . Let ; the sequence will remain bounded if the quotientconverges in the sense of tempered distribution to a pseudomeasure , as .

Remark 2. The distributional convergence in the above theorem is convergence in the Schwartz space . In other words,for all testing functions , that is, all rapidly decreasing functions.

We hereby provide the bicomplex version of Theorem 8.

Theorem 9 (bicomplex Ikehara–Korevaar theorem). Let be a bicomplex Dirichlet series where , , and with that converges to for .

Let ; then, a necessary and sufficient condition for the boundedness of is that the quotient converges in the sense of tempered distribution to a pseudomeasure , as .

Proof. Let the Dirichlet series converges to whereFor and , equivalently, , the Dirichlet series converges to the function and the Dirichlet series converges to the function
Let us denote and ; then,From Theorem 8, the necessary and sufficient condition for the boundedness of is that the quotientconverges in the sense of tempered distribution to a pseudomeasure , as .
Similarly, the necessary and sufficient condition for boundedness of is that the quotientconverges in the sense of tempered distribution to a pseudomeasure , as .
Again, by the application of the Ringleb theorem, the necessary and sufficient condition for the boundedness of is that the quotientconverges to in the sense of tempered distribution to a pseudomeasure as and .

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 sum for bicomplex Dirichlet series with hyperbolic coefficients is obtained. The conditions of convergence are affected by the 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.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

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