Fractional Derivative of the Hurwitz О
Journal of King Saud University – Science (2016) 28, 75–81
King Saud University Journal of King Saud University – Science www.ksu.edu.sa www.sciencedirect.com
ORIGINAL ARTICLE Fractional derivative of the Hurwitz f-function and chaotic decay to zero
C. Cattani a,*, E. Guariglia b a Dept. of Mathematics, University of Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy b Dept. of Physics ‘‘E. R. Caianiello’’, University of Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy
Received 26 March 2015; accepted 16 April 2015 Available online 24 April 2015
KEYWORDS Abstract In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function Fractional calculus; and Riemann zeta function is explicitly computed using the Caputo fractional derivative in the Riemann zeta function; Ortigueira sense. It is observed that the obtained results are a natural generalization of the integer Hurwitz function; order derivative. Some interesting properties of the fractional derivative of the Riemann zeta func- Dirichlet series tion are also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series. ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction fractional calculus has opened new perspectives in scientific investigation with many interesting results (Bhrawy and In recent years, fractional calculus has more and more become Abdelkawy, 2015; Liu et al., 2014; Yang et al., 2013a,b; one of the most suitable tools for research both in theory and Zhao et al., 2013). in applications, spreading over almost all fields of science and A very interesting topic is the geometrical interpretation of technology. For instance, fundamental problems of fracture fractional operators. In fact, like the integer order derivative mechanics, electromagnetism, speech signals, sound wave that gives the linear approximation of smooth (differentiable) propagation in rigid porous materials, fluid mechanics, functions, it should be expected that fractional derivative viscoelasticity, and edge detection, were easily solved and cor- might give the non-linear approximation of the local behavior rectly explained using fractional calculus (Dalir and Bashour, of non-differentiable functions (see e.g. Wang, 2012), thus 2010; Tarasov, 2008, 2006). In particular, because of the local opening new perspectives in research, by the investigation of definition of differential operators, the local approach of fractional derivatives on non-differentiable sets, like the Cantor sets (Liu et al., 2014; Yang et al., 2013a,b; Zhao et al., 2013) or by investigating the non-differentiability of * Corresponding author. E-mail addresses: [email protected] (C. Cattani), [email protected] some special functions. (E. Guariglia). Almost all special functions are functions of complex vari- Peer review under responsibility of King Saud University. ables (Abramowitz and Stegun, 1964). For this reason, the Ortiguera definition of fractional derivative of complex func- tions (Li et al., 2009; Ortigueira, 2006) seems to be the most suitable tool for studying complex functions. Using the Production and hosting by Elsevier Ortiguera fractional derivative we will give, in the following, http://dx.doi.org/10.1016/j.jksus.2015.04.003 1018-3647 ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 76 C. Cattani, E. Guariglia the explicit form of the fractional derivative of the Hurwitz and entire complex plane unless at the point s ¼ 1, where there Riemann f-function together with some properties thereof. The exists a simple pole (Guariglia, 2015) so that (Mathews and famous Riemann f-function is a special case of the more general Howell, 1997): Hurwitz f-function which is a Dirichlet series. So, we will com- limðs 1ÞfðsÞ¼1 pute first the Ortiguera fractional derivative of the Hurwitz s!1 f-function to obtain the fractional derivative of the Riemann Thus, the Riemann zeta function is a holomorphic function for f-function as a special case (for preliminary results see also any complex value s – 1. Moreover, this function is zero for Guariglia, 2015). There are several applications of the analysis the negative even integers, i.e. for s ¼ 2; 4; 6 ...(Hardy of Riemann f-function in physics, such as in the scattering and Littlewood, 1921) which are known as trivial zeroes. inverse methods where quantum potentials correspond to zer- From Eq. (2.4) there follows that there are no zeroes in the oes of the Riemann zeta (Schumayer et al., 0562; Sierra, plane regions where RðsÞ > 1, thus the only non-trivial zeroes 2010), or in the chaotic behavior of quantum energy levels must belong to the strip 0 < RðsÞ < 1, named critical strip. related to these zeroes (Pozdnyakov, 2012). To this goal we will Based on the above considerations, Hardy conjectured that show in Section 4.1 that the fractional derivative of the the zeroes not only lie on the line RðsÞ¼1 but also there exist Riemann f-function also decays chaotically to zero. In particu- 2 infinitely many non-trivial zeroes on this critical line (Beliakov lar, as application to a dynamical system we will show that zero and Matiyasevich, 2014; Hardy, 1914; Hardy and Littlewood, is an attractor for this fractional derivative. Our aim is to com- 1921; Derbyshire, 2004). pute the Caputo–Ortiguera fractional derivative of the Hurwitz zeta function and show its main properties. 2.2. Hurwitz f-function 2. Preliminary remarks on Riemann and Hurwitz zeta function The Hurwitz zeta function is defined as: 2.1. Riemann f-function X1 1 fðs; aÞ¼ ð2:5Þ ðn þ aÞs The Riemann zeta function is defined as: n¼1 X1 with RðsÞ > 1; a 2 R : 0 < a 1. Obviously, for a ¼ 1; fðs; aÞ def 1 fðsÞ¼ ðÞn 2 N; s 2 C : ð2:1Þ reduces to fðsÞ, i.e. fðs; 1Þ¼fðsÞ. Like the Riemann zeta func- ns n¼1 tion, also the Hurwitz zeta function can be extended analyti- – It is a Dirichlet series (Apostol, 2010, chp. 11) which con- cally for all complex numbers s 1; moreover in s ¼ 1it verges absolutely to an analytical function for all complex owns a simple pole (Apostol, 2010, chp. 12). numbers s such that RðsÞ > 1. This function owns an integral Also this function admits an integral representation representation (Riemann, 1859; Guariglia, 2015; Edwards and (Apostol, 2010, chp. 12): Z Riemann’s, 1974; Hardy, 1991), i.e. 1 1 e atxs 1 Z fðsÞ¼ dx 1 s 1 x 1 x CðsÞ 0 1 e s dx fð Þ¼ x ð2:2Þ CðsÞ 0 e 1 with ReðsÞ > 1. So, this function can be easily expressed in being CðsÞ the gamma function defined as: terms of the Mellin transform (Whittaker and Watson, 1927). Z The Hurwitz zeta function was first introduced in the prob- 1 s 1 x lem of analytic continuation of the Dirichlet L-function CðsÞ¼ x dx; 0 e (Mollin, 2008, 2010), i.e. a function defined by a Dirichlet L- where the integral converges absolutely if RðsÞ > 0. series. In fact, if v is a Dirichlet character modulo k, it can We recall that the Dirichlet series is defined as follows: be shown that (Apostol, 2010, p. 249): Xk r Definition 1 (Dirichlet series). A Dirichlet series is the series Łðs; vÞ¼k s vðrÞf s; k (Apostol, 2010): r¼1 so that the Hurwitz Zeta functions are a sum of L-functions X1 fðnÞ (Boyadzhiev, 2008). ð2:3Þ ns n¼1 3. Remarks on fractional calculus where fðnÞ is an arithmetical function (Apostol, 2010, chp. 1), i.e. f : N ! C and s a complex number. Let us start with the Cauchy formula of repeated integration: Clearly the Riemann zeta function is a special Dirichlet ser- Z x def 1 ies, with fðnÞ¼1. JnfðxÞ¼fð nÞðxÞ¼ ðx tÞðn 1ÞfðtÞdt Thanks to its integral representation (2.2) it can be shown ðn 1Þ! a (Cattani, 2010) that: where f is a continuous function on the interval a; b with ½ Y 1 1 a 6 x 6 b (Lazarevic´, 2014). This formula suggests us a suit- fðsÞ¼ 1 ð2:4Þ ps able generalization to a repeated integral of any real order a. p2P By replacing the factorials with Gamma function, being where P is the set of prime numbers (Hardy and Wright, 2008). Cðn 1Þ¼n!; n 2 Zþ, and the integer order n of the above The function (2.1) has a unique analytic continuation to the with some real value a 2 Rþ we have the following: Fractional derivative of the Hurwitz f-function and chaotic decay to zero 77
Definition 2 (Fractional (order) integral). Given a 2 Rþ; Using this definition, we can obtain the fractional a-deriva- t 2 R, the fractional integral of order a of a continuous tive of a complex function along the h-direction of the complex function f is the operator Ja: plane. This is a natural generalization of the Caputo deriva- Z tive, because for a real function, i.e. h ¼ p, we re-obtain (3.1) 1 t JafðtÞ¼ ðt xÞafðxÞdx: (Li et al., 2009). CðaÞ 0 4. Fractional derivatives of the Dirichlet Series and of the The fractional derivative in the Caputo sense (Caputo, Hurwitz zeta function 1967) is defined through a fractional integral, as follows:
Definition 3 (Caputo fractional derivative). The fractional In this section, the fractional derivative of a function defined derivative of order a,ofaCm-differentiable function fðxÞ is by a Dirichlet series (Apostol, 2010, chp. 11), is given. As a defined as: consequence we can easily define both the fractional derivative Z of the Hurwitz zeta function and as a special case of the m t a def m a d 1 m a 1 m Riemann zeta function. cD fðtÞ¼J m fðtÞ¼ ðt xÞ f ðxÞdx ð3:1Þ dt Cðm aÞ 0 Theorem 4.1. The fractional derivatives of the Dirichlet series with m 1 < a < m 2 Z. (2.3) and of the Hurwitz zeta function (2.5) are given by: This definition holds true for all real powers (Boyadjiev X1 a et al., 2005), including negative or null values. In fact, for a m jpða mÞ ðlog nÞ cD FðsÞ¼ð 1Þ e fðnÞ a a ns a < 0, it is D ¼ fðtÞ¼J fðtÞ while for a ¼ 0itis n¼1 0 X1 a ð4:1Þ D fðtÞ¼fðtÞ. Thus, in the fractional calculus, the two opera- ðÞlogðn þ aÞ Dafðs; aÞ¼ð 1Þmejpða mÞ tors integration and differentiation are unified as one single c ðn þ aÞs operator, sometime called differintegral (Lazarevic´, 2014; de n¼1 a Oliveira and Tenreiro, 2014). where cD ð Þ is the Ortigueira derivative (3.2),s2 C; h 2 ½½0; 2p þ As a consequence of its definition we have: and m 1 < a < m 2 Z .
Theorem 3.1. For a given a; b 2 R > 0, and any function f such Proof. Let us start with the general form of a Dirichlet series that its fractional integral exists it is: given by (2.3). Using (3.2), we have (Apostol, 2010, chp. 11): Z a b aþb b a jðp hÞða mÞ 1 m X1 (1) J J ¼ J ¼ J J a e d xejh s m a 1 a aþ1 0 cD FðsÞ¼ m fðnÞn x dx (2) J f ðxÞ¼J f ðxÞ. Cðm aÞ 0 ds n¼1 Z jðp hÞða mÞ m X1 1 e d jh ¼ fðnÞn s n xe xm a 1 dx Cðm aÞ dsm Proof. It follows directly from the definition of fractional n¼1 0 integral operator (see e.g. Kamata and Nakamula, 2002; The last integral on the RHS of this equality has been already Lazarevic´, 2014). n Zgiven in Guariglia (2015),as 1 For other properties about the differintegral defined jh n xe xm a 1 dx ¼ ejhðm aÞðlog nÞa mCðm aÞ from the Riemann–Liouville and Caputo approaches, see 0 also (Kamata and Nakamula, 2002; Lazarevic´, 2014; Bagley, where, m a > 0. There follows that: 2010). Based on the Caputo derivatives we give now a suitable generalization of fractional derivative of complex functions (Guariglia, 2015; Li et al., 2009; Ortigueira, 2006).
Definition 4 (Ortigueira fractional derivative). The Ortigueira a-derivative of a given complex function fðsÞ, is defined as (Cattani, 2010; Guariglia, 2015): Z jðp hÞða mÞ 1 m jh def e d fðxe þ sÞ Daf s Da m fðmÞ s dx c ð Þ¼ f ð Þg ¼ m a mþ1 o Cðm aÞ 0 ds x ð3:2Þ
þ a with m 1 < a < m 2 Z ; h 2 ½½0; 2p , and oD is the Ortigueira derivative operator, given by: Z "# Xm ðkÞ Since it is (Guariglia, 2015) ejðp hÞa 1 1 f ðsÞ Daf s f xejh s ejkhxk dx: o ð Þ¼ aþ1 ð þ Þ m Cð aÞ 0 x k! d k¼0 ðÞ¼ð n s 1Þmn sðÞlog n m dsm 78 C. Cattani, E. Guariglia then and, since
d s s ds ðÞ¼ ððn þ aÞ n þ aÞ logðn þ aÞ d2 n a s 1 2 n a s log n a 2 ds2 ðÞ¼ð ð þ Þ Þ ð þ Þ ðÞð þ Þ . . dm s m s m dsm ðÞ¼ð ðn þ aÞ 1Þ ðn þ aÞ ðÞlogðn þ aÞ we finally get
The computation of the fractional derivative of the Dirichlet series enables us to easily compute also the cor- responding derivative of the Hurwitz zeta function. In fact, in the previous equation, by writing fðsÞ¼fðs; aÞ, we get: Z jðp hÞða mÞ 1 m X1 a e d xejh s m a 1 cD fðs;aÞ¼ m ðn þ aÞ x dx Cðm aÞ 0 ds n¼1 Z jðp hÞða mÞ m X1 1 e d jh ¼ ðn þ aÞ s ðn þ aÞ xe Cðm aÞ dsm n¼1 0 n xm a 1dx: These two results (4.1) generalize, in a coherent way, what has been already considered in Guariglia (2015) for the frac- If we call I the last integral of the RHS, we have that: tional derivative of Riemann zeta function, so that the frac- tional derivative of the Riemann zeta function can be seen in a a more general scheme. In fact, if in cD FðsÞ we put fðnÞ¼1, a we have cD fðsÞ, which is coherent because fðsÞ is a Dirichlet series with fðnÞ¼1. Analogously for the Hurwitz zeta func- a a tion: with a ¼ 1incD fðs; aÞ, we obtain cD fðsÞ, which is coherent because fðs; 1Þ¼fðsÞ. It should be noticed that the Ortigueira fractional deriva- tive gives a natural generalization of the already known (inte- ger order) derivative of the Riemann zeta function. In fact, a þ known result is that the derivative of integer order k 2 Z0 of the Riemann zeta function is (Apostol, 2010, chp. 12)
X1 k ðlog nÞ fðkÞðsÞ¼ð 1Þk ns n¼1 which coincides with the Eqs. (4.1), by putting an integer order The last equation was obtained by changing variables twice: factor, fðsÞ¼fðs; 1Þ¼and fðsÞ¼Fðs; fðnÞ¼1Þ. first xjh ¼ z, and second z logðn þ aÞ¼x, so that: 4.1. Some properties of the fractional derivatives (4.1)
Let us consider Eq. (4.1)1 with fðnÞ¼1, i.e. X1 a ðlog nÞ DafFðsÞgðfðnÞ¼1Þ¼ð 1Þmejpða mÞ ¼ DaffðsÞg c ns c n¼1
Moreover, without restricts, we set m ¼ 1 and the constant factor ejpða mÞ equal to 1: thus, we obtain the following function
X1 a def ðlog nÞ cðs; aÞ¼ DaffðsÞgðm ¼ 1Þejpðm aÞ ¼ð 1Þ c ns n¼1 X1 a ¼ð 1Þ ½ðlog nÞn s=a : ð4:2Þ n¼1 Fractional derivative of the Hurwitz f-function and chaotic decay to zero 79
Figure 1 The real part of the complex function (4.2) with x ¼ 0 and a ¼ 0:6.
Figure 2 The imaginary part of the complex function (4.2) with x ¼ 0 and a ¼ 0:6. 80 C. Cattani, E. Guariglia
Figure 3 Parametric plot of the complex function (4.2) with x ¼ 0; a ¼ 0:6, the upper limit of the sum n ¼ 60 and the upper bound of y ¼ 30. ! In the case when the variable s is a pure real number X1 X1 X1 s=a s=a s=a s=a s ¼ x, the partial sum (for a fixed n) is a function with rapid n ¼ n 0 ¼ n decay to zero. For a pure complex number, i.e. s ¼ jy then n¼1 n¼0 n¼0 we have a complex function whose real part (4.2) R½cðs; aÞ we get from (4.2) (see Fig. 1) is a slow decay even function while the complex X1 X1 a a part I½cðs; aÞ it is a slow decay odd function (Fig. 2). cðs;aÞ¼ð 1Þ ½ðlognÞn s=a ¼ð 1Þ ½ðlogðn þ 1ÞÞn s=a If we plot the real and imaginary part of the function (4.2) n¼1 "# !n¼0 in the same plane, then we have the parametric plot of the a X1 X1 nkþ1 X1 1 complex (4.2) with x ¼ 0ofFig. 3. ¼ð 1Þ ð 1Þk nh½s h= a ; s – 0 k þ 1 h!ah We can say that the parametric plot of the fractional n¼0 k¼0 h¼0 derivatives (4.2) of the Riemann zeta function is a spiral having the origin as an asymptotic attractor. However, if we zoom in Thus, for every complex number s – 0, it is: "# around the origin we can see that the spiral becomes a self- a X1 X1 kþhþ1 intersecting non-differentiable function. k n h= a cðs; aÞ¼ð 1Þ ð 1Þ ½s k 1 h!ah Moreover, from (4.2) by the series expansions we have n¼0 h;k¼0 ð þ Þ X1 k nkþ1 There follows that, apart a constant factor, the Ortigueira logðn þ 1Þ¼ ð 1Þ kþ1 k¼0 fractional derivative of the Riemann zeta function can be also X1 expressed as a complex power series (by the Pochammer ns=a ¼ 1 nk½s k= a k!ak symbol). k¼0 being 5. Conclusion def ½s k= a ¼ sðs aÞðs 2aÞ ...ðs ðk 1ÞaÞ In this paper the integer order derivative of the Riemann zeta the so-called generalized Pochammer symbol. function has been generalized to fractional order derivative. Since it is This generalization has been obtained using the Ortigueira def- X1 X1 inition of fractional derivative of complex function which has log n ¼ logðn þ 1Þ been applied to the Hurwitz zeta function and to a Dirichlet n¼1 n¼0 series. The fractional derivative of the Riemann zeta function and, for every s – 0, being has been studied to show some interesting properties, such as Fractional derivative of the Hurwitz f-function and chaotic decay to zero 81 a chaotic decay to zero (in the Gaussian plane) and a promis- Hardy, G.H., Wright, E.M., 2008. An Introduction to the Theory of ing expression as a complex power series. The chaotic decay Numbers, 6th edition. Oxford University Press. might suggest that the fractional derivative of the Riemann Kamata, M., Nakamula, A., 2002. Riemann–Liouville integrals of zeta function is non-differentiable function around zero. fractional order and extended KP hierarchy. J. Phys. A: Math. Gen. 35, 96579670. 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