-function and chaotic decay to zero. The main purpose of this paper is generalizing Theorem A to the higher derivatives of the Riemann zeta function. It has been shown that the obtained results are a natural /FormType 1 stream removed. /Matrix [1 0 0 1 0 0] In a recent paper of the first author [5], a new connection was established between zeta functions and fractional calculus. For the AC of LfI, one can find studies in Campos’ papers and in Nishimoto’s books, where the AC is using only the Cauchy contour. In particular, its half-plane of convergence gives the possibility to better understand the $\zeta^{(\alpha)}$ and its critical strip. stream and therefore allows for the existence of some maximum, where the inequalities follow from the facts, that, Looking at our contour we notice, that as, radius of the outer circle scales by a factor of (2. and since the poles are simple, we can easily compute the residues accordingly: Now, by considering only the principal branc, Before we can get into the main part of the thesis, w, highly important relationship between the Riemann zeta function and the so-called, where the interchange of limits is being justiﬁed accordingly to the interc, integral’s boundaries stay the same, since, After doing a shift of index on the last inﬁnite series, such that it starts from. /Resources 86 0 R 109 0 obj << Once a year, the university awards this prize, financed by the Friede Springer Foundation and worth 5,000 euros, to a person who has exhibited... We study the fractional integral (fI) and fractional derivative (fD), attained by the analytic continuation (AC) of Liouville’s fI (LfI) and Riemann-Liouville fI (RLfI). In this paper, we generalize it to the higher derivatives ζ(k)(s) (k>1). © 2008-2020 ResearchGate GmbH. 0000005991 00000 n 0000001138 00000 n By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Methods in Number Theory, Proc. cu2��8�� �a�ck08Z�w���Y�ޣ� ��TU���U�SU�S5�*,�a�RU�S�C���v��*y Interested in research on Fractional Derivative? Fractional derivative of the Riemann zeta function has been explicitly computed and the convergence of the real part and imaginary parts are studied. Remember that the operation of diﬀerentiation can be iterated, meaning, rate and since the limit, if it exists, is unique, we can simplify the second deriv. derivative of Riemann zeta function and wavelet analysis is introduced. Use MathJax to format equations. What is the lowest level character that can unfailingly beat the Lost Mine of Phandelver starting encounter? 0000002891 00000 n /Length 1500 into their respective real and imaginary parts. ) Monthly 78 (1971), 645–649. endstream Euler used his product expansion of the Riemann zeta function ζ(s)=∑n=1∞1ns=∏pprime1-1ps-1,Res>1to give an analytic proof that there are infinitely many primes. /Filter /FlateDecode We cordially invite you to contribute to our open Special Issue on the topic of Fractional Cal. /Filter /FlateDecode The above operator that has been put into brackets can be interpreted as the appli-, cation of the binomial theorem, leaving us with the simpliﬁcation, where the last line followed immediately from the deﬁnition of ∆, generalized binomial theorem only allows for conv, proaches from the negative and one that approac, Note that when both the left and the right limit approach the same v, just like proving a function to be diﬀerentiable at a certain point, where w. ensure that the limit is the same from both sides. /Matrix [1 0 0 1 0 0] a classic fact of analytic number theory which states, that ev. The Riemann zeta function at 0 and 1. and will complete this chapter after being stated. for a clockwise oriented contour we get, that, , which can be expressed by the Dirichlet, ) can be derived as a direct consequence of our periodic zeta function, does not lie in our radius of convergence, ) converges absolutely and uniformely for every, ) into our second functional equation for, ) being described by its limiting process of the diﬀerence, has shown on many occasions that the deﬁnition of the fractional deriva-, We proceed by doing a direct computation of the, 1, then the fractional derivative of order, A detailed proof of this theorem can be found in Guariglia’s 2017 pap, We approach this proof by manipulating the result from Theorem 5.4 accord-, The rearranging of an inﬁnite series that we just used is only justiﬁed if a series. ) How can I label segments of a smooth curve through some nodes? By direct consideration of ~'(it) and ~(it), or what is equivalent by the functional equation, of ~'(1 +it) and $(1 +it) it follows that arg$'/~(4t) changes by There are various ways of motivating the deﬁnitions (4) and (5). Guest Editors: Mehmet Ali Özarslan, Arran Fernandez, Iván Area - Sci. through Theorem 3.9 again in the functional equation for. Derivative of the Riemann zeta function for $Re(s)>0$. >> As an application, two signal processing networks, corresponding to $\eta^{(\alpha)}$ and to its Fourier transform respectively, are shortly described. be the group of units of the ring of integers modulo, Dirichlet characters are a special class of so-called, ) be a Dirichlet series of a Dirichlet character, A non-trivial proof of this theorem can be found in, This fact can be proven using the principle of mathematical induction and the. ) However, the formula (2) cannot be applied anymore if the real part We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function. In this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the forcing of a linear elliptic fractional PDE is in one Sobolev space, then the solution is in the Sobolev space of increased order corresponding to the order of the. |h� wDn�� and the integral which needs to be justiﬁed further. 2. /Subtype /Form ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. stream /Length 15 [7, 32] been used to denote, but an entirely independent variable.). Comput. In January 2021, the University of Potsdam will award the Voltaire Prize for Tolerance, International Understanding and Respect for Differences” for the fifth time. more detail on the proofs since Apostol provided many of those only with an outline. /Length 15 Our goal is to get analytical representation of solutions for homogeneous and in-homogeneous FDEs with constant coefficients in terms of commensurate and in-commensurate fractional orders. modiﬁcation of the generalized Leibniz rule. The second definition is related to the spatial variables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. endobj Assuming the sum satisfies the Cauchy-Riemann equations, I've arrived at the following: $$\zeta'(s) = \sum_{n=1}^\infty\frac{(-1)^n2^sn^{-s}\left[\log(4)+(2^s-2)\log(n)\right]}{(2^s-2)^2},$$.