Abstract
This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz
1 Introduction
In recent years, there has been a relevant interest in fractional calculus of complex variables (see, e.g., [1,2]). Nevertheless, there are only a few papers concerning the fractional calculus of zeta functions [3,4,5]. This depends on the different unsolved problems in the theory of zeta functions. Fractional calculus of complex functions entails several problems [6], thus it has not grown as fast as real fractional calculus. Nevertheless, the fractional derivative of zeta functions is fairly easy to compute. In addition to this, fractional calculus of zeta functions has unveiled new results and applications both in dynamical system theory and signal processing [7].
The fractional derivative of the Riemann
In what follows, we deal with the fractional derivative of the Hurwitz
In this paper, we show that fractional calculus of zeta functions can provide applications in different mathematical fields, according to recent results [7,8]. In particular, we derive a functional equation for the fractional derivative of the Hurwitz
The remainder of the paper is organized into three sections. Section 2 presents some preliminaries on fractional calculus of zeta functions and Bernoulli numbers. Section 3 is devoted to analytic results on fractional calculus of zeta functions. Finally, Section 4 concludes the paper with an application in information theory.
2 Notation and background
This section is devoted to introduce some notation and definitions needed throughout the rest of the paper. Let
The Riemann
admits several generalizations. In particular, two of the most important ones are, respectively, the Hurwitz
Clearly,
and
Note that (2.3) and (2.4) hold for any
Let
Remark 2.1
The choice of the fractional derivative (2.5) depends on a relevant prerequisite. In fact, this derivative satisfies the generalized Leibniz rule:
where at least one function between
The fractional derivative (2.5) covers a fundamental role in fractional calculus of holomorphic functions. In fact, (2.6) implies [4] that
where
Accordingly,
Let
whose proof can be found in [4]. Note that (2.1) and (2.2) fulfill all hypotheses of (2.10).
Let us now recall the main properties of Bernoulli numbers with respect to zeta functions. Bernoulli polynomials
All the numbers
It is worth noting that the Riemann
Proposition 2.2
[14,16] Let
and
Furthermore, we get
Remark 2.3
From (2.11), we see at once that
Indeed, the recursive implementation of (2.13) gives
Finally, we note that (2.12) does not hold for
whose proof can be found in Appendix. Equality (2.14) is often used to show the coherence of new results with the theory of zeta functions.
3 Analytic results
In this section, we focus on analytic properties of the derivative (3.1)
It is worth noting that fractional derivatives of (2.1) and (2.2) have already been computed in [3,8] using a different definition of fractional derivative [6]. Thus, we begin by proving that the definition of fractional derivative in (2.5) gives the same results.
Theorem 3.1
Let
Proof
We proved (3.1)
Binomial series expansion gives
and so
We observe that the limit in (3.2) gives us the indeterminate form
Substituting (3.3) into (3.2) we get (3.1)
Remark 3.2
Convergence of both (3.1)
3.1 Functional equation
Theorem 3.3
Let
Proof
First, we note that this result can be proved in much the same way as [4, Theorem 9]. In fact, from (2.4), we have
We see at once that
Moreover, in view of Corollary 4 in [4], we have
and so
Therefore,
as desired.□
Remark 3.4
Let us now restrict our attention to the consistency of (3.4). Note that (2.10) yields
In particular, the Hurwitz
Of course, the proof of Theorem 3.3 can be read backwards until (3.5). Therefore, the righthand side of (3.5) converges to that of (2.4) as
Now, we are in position to characterize the functional equation (3.4). More precisely, Corollary 3.5 allows us to express (3.4) in terms of sines and cosines. Furthermore, Theorem 3.6 reduces its computational cost.
Corollary 3.5
Let
where the coefficients
Proof
Substituting the trigonometric identity
into (3.4) the proof is straightforward.□
Note that the coefficients
Theorem 3.6
Let
Proof
Let us begin by writing (2.4) in a different form. We observe that
thus the
The auxiliary function in (2.8) allows us to reduce the computational cost of (2.7) (see [5]). In the same spirit, the function
implies that (2.4) can be written as follows:
On the other hand,
From (2.6) we get
Therefore, the desired result plainly follows from (3.8) and (3.9).□
Remark 3.7
We note that Theorem 3.6 for
The author proved that (3.10) is a simplified form of (2.7). For more details we refer the reader to [5]. Hence, Theorem 3.6 coherently generalizes recent results in fractional calculus of zeta functions. Moreover, we observe that the introduction of
As in (3.7), the restriction
3.2 Integral representation
The approach proposed here is based [9] on the following representation:
where the function
is 1periodic satisfying the condition
Note that (3.11) is a direct consequence of Euler’s summation formula. Now, we are ready to give an integral representation of
Theorem 3.8
Let
where
Proof
From (3.11), we get
The proof consists in computing the three fractional derivatives in the righthand side of (3.14). First, replacing
Moreover, (2.6) implies
On the other hand,
and
Thus,
Analogously, we have
The series above reduces to only three terms since
This concludes the proof.□
It is worth noting that Apostol proved an integral representation similar to (3.13) for the integer derivative of the Hurwitz
Since
which leads to
where the last equality follows Euler’s summation formula and Stirling’s formula (see Appendix). We conclude that (3.13) reduces to (2.14) when
3.3 Link between
ζ
(
α
)
and Bernoulli numbers
In the halfplane
Of course,
is the periodic Bernoulli function [14]. Note that (3.16) is a consequence of Euler’s summation formula. For the sake of simplicity, we set
therefore, (3.16) can be rewritten as
Obviously,
which holds in the halfplane
and similarly for any
Thus, (3.20) and (3.21) imply successive closed form evaluations for the family of integrals
Theorem 3.9
Let
with
Proof
From (3.18), the linearity of
We claim that
In fact, we have
which leads us to (3.24). It remains to compute the term
Note that (3.25) holds for
Remark 3.10
We note that (3.19) differs from (3.22) in only two respects: the order of differentiation and the upper limit of second summation, where
The importance of Theorem 3.9 lies in the link between
4 An application in information theory
We conclude this paper with an application in terms of Shannon entropy. It is worth pointing out that the results of this section are based on results due to Guiasu [13].
Let
A measure of the global amount of uncertainty related to the probability distribution
Clearly
More precisely, Guiasu proved that the unique solution of the previous problem is given by
where
The proof is based on the properties of Shannon entropy [13]. Analogously, fractional calculus of zeta functions can also be used to maximize
Theorem 4.1
Let
has a solution given by
The proof of Theorem 4.1 is similar in spirit to [13], that is (4.3). Furthermore, (3.1)
As a consequence, the righthand side of (4.7) is complex. Put differently, (4.7) generalizes the integer case. In fact, let
Remark 4.2
We note that
Accordingly,
Remark 4.3
Let
Clearly, the solution in (4.6) holds for any fixed
Let us compute the values of
and, on the other hand,
that is,
Moreover,
therefore, (4.8) and (4.9) imply that
We finally note that there is a lack of study on the relation between the distribution of prime numbers and
where the symbol
Appendix
Proof of (3.15)
Proof
Note that (3.15) immediately follows from
Euler’s summation formula [9] implies
and so
Some algebraic manipulations give
Getting the limit in the last equality as
Definition (3.12) implies that
The proof is complete.□

Conflict of interest: The author declares no conflict of interest.
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