Fractional Derivative of the Riemann Zeta Function

Transcription

1 E. Guariglia Fractional Derivative of the Riemann Zeta Function Abstract: Fractional derivative of the Riemann zeta function has been explicitly computed and the convergence of the real part and imaginary parts are studied. In particular, this fractional derivative is equal to a complex series, whose real and imaginary parts are convergent in a half-plane which depends on the order of the fractional derivative. Keywords: Fractional derivative; Riemann zeta function; gamma function; convergence. 1 Introduction It is well known that Riemann zeta function plays an important role both in number theory and in several applications of quantum mechanics: in particular, the Riemann hypothesis can be formulated using the quantum terminology (Pozdnyakov 212); furthermore, its zeros are related with the Hamiltonian of a quantum mechanical system (Sierra 21). Fractional calculus and fractional derivatives have been increasingly studied due to the many practical and theoretical applications: especially its geometrical interpretation gives us a powerful mathematical tool to model the most advanced concepts of modern physics (Adda 1997, Podlubny 22, Wang 212). Several authors have obtained the fractional derivative of complex functions, in particular in (Owa 28) it is shown how to extend the fractional derivative to analytical functions on the unit circle U = z C : z < 1 as a complex series f (z) = z + a n z n (1) This possibility seems to be unaffordable for the Riemann ζ -function, because this function can not be expressed as (1). In the following chapter, we will approach this problem by using a suitable modification of the Caputo fractional derivative as given by Ortigueira. It is known that the main advantage of the Caputo derivative is that the fractional derivative of a constant is zero, which is not true for the Riemann- Liouville derivative. In his paper (Ortigueira 26), Ortigueira defines a generalized Caputo derivative for complex functions with respect to a given direction of the complex plane. Thus the Ortigueira fractional derivative can be considered as the direc- n=2 E. Guariglia: Dept. of Physics "E. R. Caianiello", University of Salerno, Via Giovanni Paolo II, 8484 Fisciano, Italy, eguariglia@unisa.it 215 E. Guariglia This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3. License. Download Date 2/8/18 8:2 PM

2 358 E. Guariglia tional Caputo fractional derivative, so that the Ortigueira fractional derivative gives us the fractional derivative of a complex function in the θ-direction of the complex plane. It will be shown that Ortigueira-Caputo fractional derivative of the Riemann ζ -function can be easily computed as a complex series. The real and imaginary parts of this complex series will be studied: in particular it will be shown where these two series converge. 2 Preliminary Remarks on Riemann Zeta Function The Riemann zeta function is defined as (Whittaker et al. 1927): ζ (s) 1, (n N, s C) (2) ns This infinite series (Cattani 21, Voxman et al. 1981) converges for all the complex numbers such that R(s) > 1. This function can be also defined by the integral: where Γ is the gamma function (Apostol 21), given by: ζ (s) = 1 x s 1 Γ(s) e x dx (3) 1 Γ(t) x t 1 dx (4) e x in which t C and the last integral converges if R(t) > (Abramowitz et al. 1965). The Riemann zeta function can be defined only for R(s) > 1, otherwise the absolute value of n s would be too small and the series in (2) would diverge. Riemann has proven that the function ζ defined by this series has an unique analytical continuation to the entire complex plane, excluding the point s = 1, where this function has a simple pole with complex residue 1 (Riemann 1859). Moreover, it can be shown that: lim(s 1)ζ (s) = 1 (5) s 1 so that Riemann zeta function is meromorphic on the whole complex plane, and holomorphic for every complex s except for s = 1. Concerning its zeros, ζ is equal to zero at the negative even integers, i.e. ζ (s) = when s is 2, 4, 6,...: they are called trivial zeros (Edwards 1974, Voxman et al. 1981). Euler has also shown that (Hardy 1914, Mollin 21): ζ (s) = 1 1 p s (6) p P Download Date 2/8/18 8:2 PM

3 Fractional Derivative of the Riemann Zeta Function 359 where P is the set of prime numbers (Mathai et al. 28, Davenport 28). Moreover, by using the last equation, we can easily see that there are no zeros in the plane regions where R(s) > 1, so that the only non-trivial zeros must belong to the strip < Re(s) < 1, also called critical strip (Hardy et al. 1921). Riemann conjectured that they lie on the line R(s) = 1/2, called critical line: this is the famous Riemann hypothesis. In 1914, Hardy has shown that ζ has infinitely many non-trivial zeros on the critical line (Hardy et al. 28). 3 Derivative in the Ortigueira Sense and Caputo Derivative in Complex Plane Recently, M. D. Ortigueira has proposed a new type of fractional derivative, defined in the complex plane, starting from the following generalization of the Cauchy s integral formula (Ortigueira 211): D α f (s) = Γ(α + 1) 2πj C f (w) dw (7) (w s) α+1 where C is a U shaped contour that encircles the branch cut line, while j is the imaginary unit. This operator (called the derivative in the Ortigueira sense) gives us the fractional derivative in the θ-direction of the complex plane: it is generally indicated by O D α and is defined as follows: OD α f (s) = ej(π θ)α Γ( α) f (xe jθ + s) n k= f (k) (s) k! e jkθ x k x α+1 dx (8) in which n = α is the largest integer non greater than α, while s belongs to the complex plane and θ [, 2π). The derivative in the Ortigueira sense has many proprieties: below we discuss three of the most important (Li et al. 29, Ortigueira et al. 214). 1. Consistency with order-derivative If n 1 < α < n Z +, f (s) is analytic in a region that contains the Hankel contour C, and without loss of generality we can suppose θ [, 2π): it is easy to show (Li et al. 29) that: lim α n O D α f (s) = f (n) θ (s) lim α (n 1) O D α f (s) = f (n 1) (s) + θ Download Date 2/8/18 8:2 PM

4 36 E. Guariglia where f n θ is the n-th directional derivative in θ-direction. 2. Composition with integer-order derivatives Let A k be the set of complex functions f : D C C such that d k ds k f (xe j θ + s)dx = d k ds k f (xej θ + s)dx in which θ [, 2π) and k Z +. If f A k, in (Li et al. 29) it is shown that: d k ( ds k OD α f (s) ) ( ) = O D α d k ds k f (s) 3. Composition with itself Let us give k < α < k + 1 Z +, h < β < h + 1 Z + and f A k. The following equations hold true: a) O D α f (s) ( OD α f (s) ) = O D α ( OD α f (s) ) ( ) b) O D α f (s) OD β f (s) = O D β ( OD α f (s) ) ( ) c) O D α f (s) OD β f (s) = O D β ( OD α f (s) ) O D α+β f (s) where the one holds for f A max[h,k] (Li et al. 29). We recall that the classical Caputo fractional derivative of order α, of a C m -differentiable function f (t) is defined as (Bagley 27): CD α f (t) 1 Γ(m α) t (t x) m α 1 f m (x) dx (9) with m 1 < α < m Z. Fractional derivatives in Ortigueira sense have many other proprieties (Ortigueira 26), but for our purposes the most significant fact is how it generalizes the classical Caputo derivative (Caputo 1967, De Oliveira et al. 214) in the complex plane. By using the Ortigueira derivative, the α-th Caputo derivative along with the θ-direction of the complex plane is defined as: ( ) CD α f (s) O D α m f (m) (s) = ej(π θ)(α m) Γ(m α) f (m) (xe jθ + s) x α m+1 dx (1) where m 1 < α < m Z + (Kilbas et al. 26, Li et al. 29): this operator is called Ortigueira-Caputo fractional derivative of the complex function f (s). Download Date 2/8/18 8:2 PM

5 Fractional Derivative of the Riemann Zeta Function Fractional Derivative of ζ(s) We now have all the mathematical tools necessary to calculate the Ortigueira-Caputo fractional derivative of the Riemann ζ -function. From (1), it follows that: CD α ζ (s) = ej(π θ)(α m) Γ(m α) = ej(π θ)(α m) Γ(m α) d m ( ds m d m ds m ζ ( xe jθ + s )) 1 x α m+1 dx ( ) n s n xejθ x m α 1 dx Bringing the integral sign under both derivative and summation, we obtain: CD α ζ (s) = ej(π θ)(α m) Γ(m α) d m ds m n s n xejθ x m α 1 dx (11) Let us begin by computing the integral in the right hand side (RHS) of the equation 11: by a change of variables xe jθ = z, we have: n xejθ x m α 1 dx = n z e jθ(m α 1) z m α 1 e jθ dz = e jθ(m α) = e jθ(m α) e z log n z m α 1 dz e x x m α 1 (log n) 1+α m dx log n = e jθ(m α) (log n) α m Γ(m α) where we have performed another change of variable, i.e. z log n = x; furthermore, we have used the definition of gamma function (see equation 4). Given that, from the definition of Ortigueira-Caputo fractional derivative m α >, the last RHS of the continued equality above makes sense. Substituting: CD α ζ (s) = ej(π θ)(α m) d m Γ(m α) ds m n s e θ(m α) ( log n) ) α m Γ(m α) = e jπ(α m) (log n) α m d m ( n s ) ds m (12) Download Date 2/8/18 8:2 PM

6 362 E. Guariglia We are now ready to compute the derivative in the RHS: so that d ( n s ) = n s log n ds d 2 ( n s ) ds 2 = ( 1) 2 n s (log n) 2. d m ds m ( n s ) = ( 1) m n s (log n) m CD α ζ (s) = ( 1) m e jπ(α m) = ( 1) m e jπ(α m) (log n) α m n s (log n) m (log n) α n s i.e. CD α ζ (s) = ( 1) m e jπ(α m) (log n) α n s (13) In Fig. 4 is represented the modulus of this fractional derivative, with α =.5 and upper limit of sum n = 1. For simplicity, from now on, we will denote the fractional derivative computed above with ψ: ψ C D α ζ in which m 1 < α < m Z + and ζ = ζ (s), it follows immediately that ψ = ψ(α, s). Let us write this function in rectangular form (Mathews et al. 28). Since s C s = x + jy, it is: from which we obtain: n s = n x n jy = n x e jy log n = n x( cos(y log n) j sin(y log n) ) ψ(α, s = x + jy) = ( 1) m ( cos(π(α m)) + j sin(π(α m)) ) (log n) α n x( cos(y log n) j sin(y log n) ) = ( 1) m (log n) α n x( cos(π(α m)) cos(y log n) + sin(π(α m)) sin(y log n) + j ( sin(π(α m)) cos(y log n) cos(π(α m)) sin(y log n) )) Download Date 2/8/18 8:2 PM

7 Fractional Derivative of the Riemann Zeta Function 363 It follows that: R(ψ(α, s)) = ( 1) m and similarly: I(ψ(α, s)) = ( 1) m (log n) α n x( ) cos(π(α m)) cos(y log n) + sin(π(α m)) sin(y log n) = ( 1) m (log n) α n x 1 ( cos(π(α m) + y log n) + cos(π(α m) y log n) 2 ) + cos(π(α m) y log n) cos(π(α m) + y log n) = ( 1) m (log n) α n x cos(π(α m) y log n) (log n) α n x( ) sin(π(α m)) cos(y log n) cos(π(α m)) sin(y log n) = ( 1) m (log n) α n x 1 ( sin(π(α m) + y log n) + sin(π(α m) y log n) 2 ) sin(π(α m) + y log n) sin(y log n π(α m)) = ( 1) m (log n) α n x sin(π(α m) y log n) In summary, we have shown the following property. Theorem 37. The Ortigueira-Caputo fractional derivative of the Riemann ζ -function is equal to: CD α ζ (s) = ( 1) m e jπ(α m) (log n) α n s (14) and its real and imaginary parts are given by: R(ψ(α, s)) = ( 1) m I(ψ(α, s)) = ( 1) m (log n) α n x cos(π(α m) y log n) (log n) α n x sin(π(α m) y log n) (15) where s C and m 1 < α < m Z +. 5 Convergence of R(ψ(α, s)) and I(ψ(α, s)) At this point we check whether R(ψ(α, s)) and I(ψ(α, s)) converge somewhere, using the classical comparison test and a generalization of the well known harmonic series. Download Date 2/8/18 8:2 PM

8 364 E. Guariglia Theorem 38. (Comparison test) Suppose that converges and b n a n. Under these hypotheses, the series converges. a n b n Proof. See (Stirling 29, pp. 99-1) Corollary 6. Let p R, the following series converges if p > 1 and diverges if p 1. 1 n p (16) Proof. Just use the well known property about the convergence of the harmonic series. The series 16 is called p-series (Marsden et al. 1993) and if p > 1 it is equal to ζ (p). These two properties are essential to show the following statement. Theorem 39. The two real functions R(ψ(α, s)) and I(ψ(α, s)) converge in the halfplane: Proof. Since the sin( ) is a bounded function in [ 1, 1], it is: x > 1 + α (17) sin(x) sin(x) 1 therefore: (log n) α (log n)α n x sin(π(α m) y log n) n x < nα n x = 1 n x α In the last RHS of the relations above, the last term form is that of a p-series: it converges for x α > 1 Using the comparison test, it is clear that I(ψ(α, s)) also converges in the halfplane x > 1 + α. Similarly, it can be shown that the same holds true for R(ψ(α, s)) (see fig. 1). Download Date 2/8/18 8:2 PM

9 Fractional Derivative of the Riemann Zeta Function 365 Fig. 1. Half-plane of convergence for R(ψ(α, s)), I(ψ(α, s)) and ψ(α, s): it is interesting to note that ζ(s) converges for x > 1 and its α-order fractional derivative converges for x > 1 + α. There follows that Corollary 7. The fractional derivative of the Riemann zeta-function converges in the half-plane (17). Proof. It follows from the theorem above remembering that a complex series converges if and only if both the real and imaginary parts converge (Lang 23). Download Date 2/8/18 8:2 PM

10 366 E. Guariglia Fig. 2. 3D plot of the function R(ψ(α, s)) with α =.5, upper limit of the sum n = 1 and (x, y) [ 1, 1] 2 : the surface satisfies the (17). In Fig. 2 (respectively Fig. 3 and Fig. 4), it is plotted R(ψ(α, s)) (respectively I(ψ(α, s)) and ψ(α, s) ) with α =.5 and upper bound n = 1: the surface behavior is in accord with the above theorem. Conclusion In this chapter, the fractional derivative of the Riemann zeta function was computed, using the Caputo derivative in the Ortigueira sense. It was shown where the real and imaginary parts of this new function converge. Bibliography Abramowitz et al. (1965) Abramowitz, M. & Stegun, I. A. (1964). Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington DC, Dover Pubblications. Adda, F. B. (1997). Geometric interpretation of the fractional derivative. Journal of Fractional Calculus, 11, Download Date 2/8/18 8:2 PM

11 Fractional Derivative of the Riemann Zeta Function 367 Fig. 3. 3D plot of the function I(ψ(α, s)) with α =.5, upper limit of the sum n = 1 and (x, y) [ 1, 1] 2 : also here, the condition 17 is true. Fig. 4. 3D plot of the function ψ(α, s) with α =.5, upper limit of the sum n = 1 and (x, y) [ 1, 1] 2 : it still satisfies the (17) and has the same shape of ζ(s) (see Pegg et al. 24). Apostol, T. M. (21). Introduction to Analytic Number Theory. New York: Springer. Bagley, R. (27). On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials. Fractional Calculus and Applied Analysis, 1 (2), Download Date 2/8/18 8:2 PM

12 368 E. Guariglia Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent-II. Geophysical Journal of the Royal Astronomic Society, 13 (5), Cattani, C. (21). Fractal Patterns in Prime Numbers Distribution. Computational Science and its Applications. Berlin/Heidelberg Springer, 617, Davenport, H. (28). The Higher Arithmetic: An Introduction to the Theory of Numbers. Cambridge: Cambridge University Press. De Oliveira, E. C. & Tenreiro Machado, J. A. (214). A Review of Definitions for Fractional Derivatives and Integrals. Mathematical Problems in Engineering, 6 pages. Edwards, H. M. (1974). Riemann s Zeta Function. New York: Academic Press. Hardy, G. H. (1914). Sur les zeros de la fonction ζ(s) de Riemann. Comptes Rendus de l Académie des Sciences, 158, Hardy, G. H. & Littlewood, J. E. (1921). The zeros of Riemann s zeta-function on the critical line. Mathematische Zeitschrift, 1 (3-4), Hardy, G. H. & Wright, E. M. (28). An Introduction to the Theory of Numbers. Oxford: Oxford University Press. Kilbas, A. A., Srivastava, H. M. & Trujillo, J. J. (26). Theory and Applications of Fractional Differential Equations. New Jersey: Springer. Lang, S. (23). Complex Analysis. New York: Springer. Li, C., Dao, X. & Guo, P. (29). Fractional derivatives in complex planes. Nonlinear Analysis: Theory, Methods & Applications, 71, Marsden, J. E. & Hoffman, M. J. (22). Elementary Classical Analysis. San Francisco: W. H. Freeman and Company. Mathai, A. M. & Haubold, H. J. (28). Special function for Applied Scientists. New York: Springer. Mathews, J. H. & Howell, R. W. (28). Complex Analysis for Mathematics and Engineering. New York: Springer-Verlag. Mollin, R. A. (21). Advanced Number Theory with Applications. Boca Raton: CRC Press. Ortigueira, M. D. (26). A coherent approach to non-integer order derivatives. Signal Processing, 86 (1), Ortigueira, M. D. (211). Fractional Calculus for Scientists and Engineers. London: Springer. Ortigueira, M. D. & Tenreiro Machado, J. A. (214). What is a fractional derivative?. Journal of Computational Physics, 293, Owa, S. (28). Some properties of fractional calculus operators for certain analytic functions. Proceedings of International Symposium on New Development of Geometric Function Theory and its Applications, Bangi (Malaysia), Pegg, E. Jr. & Weisstein, E. W. (24). Seven Mathematical Tidbits. MathWorld Headline News. Podlubny, I. (22). Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. Fractional Calculus and Applied Analysis, 5 (4), 22, Pozdnyakov, D. (212). Physical interpretation of the Riemann hypothesis. arxiv: Riemann, G. F. B. (1859). Ueber die Anzahl der Primzahlen unter einergegebenen Grösse. Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1 pages. Sierra, G. (21). A physics pathway to the Riemann hypothesis. arxiv:math-ph/ Stirling, D. S. G. (29). Mathematical Analysis and Proof. Philadelphia, Woodhead Publishing. Voxman, W. L. & Goetschel, R. H. (1981). Advanced Calculus: An Introduction to Modern Analysis. New York: CRC Press. Wang, X. (212). Fractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering. Proceedings of The Fifth Symposium on Fractional Differentiation and its Applications, Hohai University (China). Whittaket, T. & Watson, G. N. (1927). A Course of Modern Analysis. Cambridge: Cambridge University Press. Download Date 2/8/18 8:2 PM