Fractional calculus , zeta functions and Shannon entropy

Abstract: This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.


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 ζ function allows several generalizations. These include, inter alia, the fractional derivative of the Hurwitz ζ function and that of a Dirichlet series [8]. Following the approach of Apostol [9], this paper follows up on fractional analysis of zeta functions. Further studies on the link between fractional calculus of Riemann zeta function and the distribution of prime numbers can be found in [5,7].
In what follows, we deal with the fractional derivative of the Hurwitz ζ function computing the functional equation. For the fractional derivative of the Riemann ζ function and the relative functional equation, we refer the reader to [4,5]. More precisely, the generalized Leibniz rule enables the computation of the functional equation sought. Thus, the fractional derivative of zeta functions seems to play the same relevant role in fractional calculus that the class of zeta functions plays in pure and applied mathematics. In addition to this, zeta functions make the definition of a probability distribution on allowable. Moreover, the nontrivial zeros of the Riemann zeta function are nowadays one of the most relevant unsolved problems in mathematics. The distribution of these zeros seems to obey to no mathematical law. Several attempts to prove the Riemann hypothesis have failed in recent years. Quite recently, some analytic results shed new light on the Riemann hypothesis [10]. On the other hand, random matrix theory is a fundamental tool in modern number theory and, more precisely, in the class of zeta functions. For more details we refer the reader to [11,12]. This fact suggests to apply the theory of zeta functions to statistical problems (e.g., Shannon entropy [13]).
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 ζ function. Likewise, we give an integral representation of this derivative. Moreover, we prove a connection between fractional calculus of zeta functions and Bernoulli numbers. Finally, we show that our analytic results find application in the concept of Shannon entropy.
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.

Notation and background
This section is devoted to introduce some notation and definitions needed throughout the rest of the paper. Let ∈ α and ∈ n 0 with = ∪ { } 0 0 . We use the notation α n to denote the nth falling factorial of α [4]. Moreover, ⌊ ⌋ α stands for the integer part of a real number α. Here and subsequently, s denotes a complex variable.
The 1 . Fractional calculus of Dirichlet series can be found in [8] and this topic exceeds the scope of this paper. It is worth noting that (2.1) and (2.2)1 exhibit the same analytic behavior. In fact, these zeta functions either converge in the half-plane > s Re 1. Furthermore, (2.1) and (2.2)1 own a unique analytical continuation to the entire complex plane, except a simple pole (with residue 1) in = s 1, given [14] by  where at least one function between f and g in (2.6) is analytic in the region ⊆ D . A proof of (2.6) can be found in [4]. where . Note that the functional equation (2.7) can easily be written in terms of sines and cosines and entails high computational cost. The author dealt with these problems in [5]. In particular, the method proposed in [5] reduces the computational cost of (2.7) to only one infinite series. This technique involves the auxiliary function ψ defined as follows: be a function such that the fractional incremental ratio in (2.5) is uniformly convergent in D. Then, for any ∈ ⧹ 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 ( ) B s n of the complex variable s are defined by the following equation: It is worth noting that the Riemann ζ function is closely linked to Bernoulli numbers, as next result points out. Furthermore, we get The other Bernoulli numbers are given by the following expansion: Indeed, the recursive implementation of (2.13) gives , etc. Moreover, the functional equation (2.3) gives no information on ( + ) ζ n 2 1 (both members vanish). In the current literature, no simple formula for positive odd values of ζ is known (see, e.g., [14,17]).
Finally, we note that (2.12) does not hold for = n 0. Thus, we recall that (2.14) 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.

Analytic results
In this section, we focus on analytic properties of the derivative (3.1) 2 . In particular, we state and prove the functional equation. Moreover, we give an integral representation of (3.1)1 in terms of Bernoulli numbers. 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.
Proof. We proved (3.1)1 in [4]. The rest of the proof can be handled in much the same way. Indeed, let us prove The proof of (3.1) 2 is similar and we thus leave it to the reader. □ Remark 3.2. Convergence of both (3.1)1 and (3.1) 2 depends on α (see [3,8]). Accordingly, Theorem 3.1 implies that convergence of (3.1) 3 depends on both α and f .

Functional equation
We see at once that   In particular, the Hurwitz ζ function fulfills all hypotheses of (2.10), and thus Of course, the proof of Theorem 3.3 can be read backwards until (3.5). Therefore, the right-hand side of (3.5) converges to that of (2.4) as → + α 0 . As a consequence, The auxiliary function in (2.8) allows us to reduce the computational cost of (2.7) (see [5]). In the same spirit, the function ψ q defined by implies that (2.4) can be written as follows: On the other hand, Therefore, the desired result plainly follows from (3.8) and (3.9). □ Remark 3.7. We note that Theorem 3.6 for = = p q 1 gives 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 ψ q relies on the fact that :R e 0 .

Integral representation
The approach proposed here is based [9] on the following representation: where the function φ 2 defined by is 1-periodic 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 ( ) , as next result points out. where φ 2 is defined by (3.12).
Proof. From (3.11), we get 14) The proof consists in computing the three fractional derivatives in the right-hand side of (3.14). First, replacing n with a in (3.1)1 gives    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 = s 0, = a 1 and → + α 1 . Accordingly, Theorem 3.8 is consistent with the theory of zeta functions.

Link between ( ) ζ α and Bernoulli numbers
In the half-plane > − s n Re 2 with ∈ n , the Riemann ζ function can be expressed as [9] follows: is the periodic Bernoulli function [14]. Note that The importance of Theorem 3.9 lies in the link between ( ) ζ α and Bernoulli numbers. Similar to Theorem 3.8, the consistency of (3.22) plainly follows from (3.19).