# 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
*inter alia*, the fractional derivative of the Hurwitz

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

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:

The fractional derivative (2.5) covers a fundamental role in fractional calculus of holomorphic functions. In fact, (2.6) implies [4] that

Let

Let us now recall the main properties of Bernoulli numbers with respect to zeta functions. Bernoulli polynomials

## Proposition 2.2

[14,16] *Let*
*The values of the Riemann*
*function for non-positive integers and positive even numbers are given, respectively, by*

*and*

*Furthermore, we get*

## Remark 2.3

From (2.11), we see at once that

Finally, we note that (2.12) does not hold for

## 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*
*and*
*Then*

## Proof

We proved (3.1)

## Remark 3.2

Convergence of both (3.1)

### 3.1 Functional equation

### Theorem 3.3

*Let*
*and*
*Then, for any*

### 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

### Remark 3.4

Let us now restrict our attention to the consistency of (3.4). Note that (2.10) yields

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*
*and*
*Then, for any*

*where the coefficients*

*and*

*are given by*

### Proof

Substituting the trigonometric identity

Note that the coefficients
*s*, as in the case of integer order [9].

### Theorem 3.6

*Let*
*and*
*Then, for any*

### Proof

Let us begin by writing (2.4) in a different form. We observe that

### Remark 3.7

We note that Theorem 3.6 for

### 3.2 Integral representation

The approach proposed here is based [9] on the following representation:

### Theorem 3.8

*Let*
*and*
*In the half-plane*
*we have*

*where*

*is defined by*( 3.12).

### Proof

From (3.11), we get

It is worth noting that Apostol proved an integral representation similar to (3.13) for the integer derivative of the Hurwitz

### 3.3 Link between
ζ
(
α
)
and Bernoulli numbers

In the half-plane

### Theorem 3.9

*Let*
*In the half-plane*
*with*
*we have*

*with*

*and*

*as in*( 3.17).

### Proof

From (3.18), the linearity of

### 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

## Theorem 4.1

*Let*
*The maximization of Shannon entropy*
*subject to the constraints in* (4.1) *and*

*has a solution given by*

The proof of Theorem 4.1 is similar in spirit to [13], that is (4.3). Furthermore, (3.1)

## Remark 4.2

We note that

## Remark 4.3

Let

Clearly, the solution in (4.6) holds for any fixed

## Appendix

### Proof of (3.15)

### Proof

Note that (3.15) immediately follows from

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

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**Received:**2020-08-11

**Revised:**2020-12-06

**Accepted:**2021-01-11

**Published Online:**2021-04-26

© 2021 Emanuel Guariglia, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.