This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz 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.
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 , 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 .
The fractional derivative of the Riemann function allows several generalizations. These include, inter alia, the fractional derivative of the Hurwitz function and that of a Dirichlet series . Following the approach of Apostol , 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 function computing the functional equation. For the fractional derivative of the Riemann 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 . 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 ).
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 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.
2 Notation and background
This section is devoted to introduce some notation and definitions needed throughout the rest of the paper. Let and with . We use the notation to denote the th falling factorial of . Moreover, stands for the integer part of a real number . Here and subsequently, denotes a complex variable.
The Riemann function defined by
admits several generalizations. In particular, two of the most important ones are, respectively, the Hurwitz function and Dirichlet series defined by
Clearly, and . Fractional calculus of Dirichlet series can be found in  and this topic exceeds the scope of this paper. It is worth noting that (2.1) and (2.2) exhibit the same analytic behavior. In fact, these zeta functions either converge in the half-plane . Furthermore, (2.1) and (2.2) own a unique analytical continuation to the entire complex plane, except a simple pole (with residue 1) in , given  by
Let be a function analytic inside the region and continuous on its contour . The forward Grünwald-Letnikov fractional derivative of is defined as follows:
The choice of the fractional derivative (2.5) depends on a relevant prerequisite. In fact, this derivative satisfies the generalized Leibniz rule:
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 . In particular, the method proposed in  reduces the computational cost of (2.7) to only one infinite series. This technique involves the auxiliary function defined as follows:
Let be a function such that the fractional incremental ratio in (2.5) is uniformly convergent in . Then, for any , we have
Let us now recall the main properties of Bernoulli numbers with respect to zeta functions. Bernoulli polynomials of the complex variable are defined by the following equation:
All the numbers are called Bernoulli numbers and simply denoted by . Therefore,
It is worth noting that the Riemann function is closely linked to Bernoulli numbers, as next result points out.
Furthermore, we get
From (2.11), we see at once that and . 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 (both members vanish). In the current literature, no simple formula for positive odd values of is known (see, e.g., [14,17]).
Finally, we note that (2.12) does not hold for . Thus, we recall that
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) . In particular, we state and prove the functional equation. Moreover, we give an integral representation of (3.1) 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 . Thus, we begin by proving that the definition of fractional derivative in (2.5) gives the same results.
Let and . Then
Binomial series expansion gives
We observe that the limit in (3.2) gives us the indeterminate form . L’Hôpital’s rule implies that
3.1 Functional equation
Let and . Then, for any ,
We see at once that
Moreover, in view of Corollary 4 in , we have
In particular, the Hurwitz 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 . As a consequence, (3.4) is consistent with the theory of zeta functions.
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.
Let and . Then, for any ,
where the coefficients and are given by
Substituting the trigonometric identity
into (3.4) the proof is straightforward.□
Let , and . Then, for any ,
Let us begin by writing (2.4) in a different form. We observe that
thus the -periodicity of the complex exponential implies
implies that (2.4) can be written as follows:
On the other hand,
From (2.6) we get
We note that Theorem 3.6 for gives
The author proved that (3.10) is a simplified form of (2.7). For more details we refer the reader to . Hence, Theorem 3.6 coherently generalizes recent results in fractional calculus of zeta functions. Moreover, we observe that the introduction of relies on the fact that
3.2 Integral representation
The approach proposed here is based  on the following representation:
where the function 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.
Let and . In the half-plane , we have
where is defined by (3.12).
From (3.11), we get
Moreover, (2.6) implies
On the other hand,
Analogously, we have
The series above reduces to only three terms since for any . As a consequence,
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 function . Thus, Theorem 3.8 gives a fractional generalization of this result. Now, we are in position to deal with the consistency of (3.13). To check this, we observe that
Since , we have
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 , and . Accordingly, Theorem 3.8 is consistent with the theory of zeta functions.
3.3 Link between and Bernoulli numbers
In the half-plane with , the Riemann function can be expressed as  follows:
Of course, are Bernoulli numbers and
therefore, (3.16) can be rewritten as
which holds in the half-plane with . In the special case we get
and similarly for any it follows
Thus, (3.20) and (3.21) imply successive closed form evaluations for the family of integrals . For more details, we refer the reader to . Now, we are able to generalize (3.19) for a fractional order of differentiation.
Let . In the half-plane with , we have
with and as in (3.17).
From (3.18), the linearity of implies that
We claim that
In fact, we have
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 .
Let be a probability distribution on , i.e. for any and
A measure of the global amount of uncertainty related to the probability distribution is given by Shannon entropy defined as follows:
Clearly . Many modern techniques in number theory are based on statistics. In particular, nontrivial zeros of the Riemann function can be viewed as a statistical distribution (see, e.g., [18,19]). The Riemann function can be used to maximize subject to constraints (4.1) and
More precisely, Guiasu proved that the unique solution of the previous problem is given by
where is the unique real number such that and
The proof is based on the properties of Shannon entropy . Analogously, fractional calculus of zeta functions can also be used to maximize , as stated below.
Let . The maximization of Shannon entropy subject to the constraints in (4.1) and
has a solution given by
As a consequence, the right-hand side of (4.7) is complex. Put differently, (4.7) generalizes the integer case. In fact, let . Replacing by , the right-hand side of (4.7) gives the derivative . Being , the integer derivative of is real. Conversely, the fractional derivative of real-valued functions can also be complex.
We note that implies that in (4.4). Moreover,
Let be the operator of statistical mean with respect to . It is worth noting that . Likewise, .
Clearly, the solution in (4.6) holds for any fixed . Note that Theorem 4.1 does not prove uniqueness of the solution (4.6). This depends on the lack of a relation equivalent to (4.4) for . We recall that uniqueness of (4.3) follows  from
and, on the other hand,
We finally note that there is a lack of study on the relation between the distribution of prime numbers and . The author proved  that
where the symbol means that both sides above converge or diverge together. Thus, fractional calculus of zeta functions and classical theory of zeta functions seem to have similar behaviors with respect to the prime distribution.
Proof of (3.15)
Note that (3.15) immediately follows from
Euler’s summation formula  implies
Some algebraic manipulations give
Getting the limit in the last equality as , we have
Definition (3.12) implies that , thus the improper integral above converges absolutely. Moreover, Stirling’s formula entails that
The proof is complete.□
Conflict of interest: The author declares no conflict of interest.
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