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On the Critical Strip of the Riemann zeta Fractional Derivative

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Języki publikacji
EN
Abstrakty
EN
The α-order fractional derivative of the Dirichlet η function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function ζ(α) on the critical strip. The convergence of η(α) is studied. In particular, its half-plane of convergence gives the possibility to better understand the ζ(α) and its critical strip. As an application, two signal processing networks, corresponding to η(α) and to its Fourier transform respectively, are shortly described.
Wydawca
Rocznik
Strony
459--472
Opis fizyczny
Bibliogr. 20 poz., rys., wykr.
Twórcy
autor
  • Engineering School (DEIM), University of Tuscia, Largo dell’Università, 01100 Viterbo, Italy
autor
  • Dept. of Physics “E. R. Caianiello”, University of Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy
autor
  • School of Electronic Science and Engineering, Nanjing University, Nanjing, Jiangsu 210046, China
Bibliografia
  • [1] Adams RW. A Signal-processing Interpretation of the Riemann Zeta Function. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Philadelphia, PA, USA, 19-23 March 2005, Volume 4, p. 77-80. doi: 10.1109/ICASSP.2005.1415949.
  • [2] Agarwal P, Cho I, Paris RB. Extended Riemann-Liouville Fractional Derivative Operator and its Applications. Journal of Nonlinear Science and Applications. 2015; 8: 451-466. URL http://hdl.handle.net/10373/2180.
  • [3] Agarwal P, Dragomir SS, Park J, Jain S. q-Integral inequalities associated with some fractional q-integral operators. Journal of Inequalities and Applications. 2015; 345: 1-13. doi: 10.1186/s13660-015-0860-8.
  • [4] Agarwal RP, Agarwal P. Extended Caputo fractional derivative operator. Advanced Studies in Contemporary Mathematics. 2015; 25: 301-316.
  • [5] Agarwal P, Nieto JJ. Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Mathematics. 2015; 13: 537-546. doi: 10.1515/math-2015-0051.
  • [6] Apostol T. Introduction to Analytic Number Theory. Springer; 1998. ISBN-13: 978-0387901633, 10: 0387901639.
  • [7] Beerends RJ, ter Morsche HG, van den Berg JC, van de Vrie EM. Fourier and Laplace Transforms. Cambridge University Press; 2003. ISBN-13: 9780521806893, 10: 0521806895.
  • [8] Borwwein P, Choi S, Rooney B, Weirathmueller A. The Riemann Hypothesis: A Resourse for the Afficionado and Virtuoso Alike. Springer; 2007. ISBN-13: 978-0387721255, 10: 0387721258.
  • [9] Cattani C, Guariglia E. Fractional derivative of Hurwitz ζ -function. Journal of King Saud University - Science. 2016; 28: 75-81. URL http://dx.doi.org/10.1016/j.jksus.2015.04.003.
  • [10] Edwards HM. Riemanns Zeta Function. Academic Press; 1974. ISBN-10: 0486417409, 13: 978-0486417400.
  • [11] Guariglia E. Fractional Derivative of the Riemann Zeta Function. In: Fractional Dynamics; Cattani C, Srivastava H, Yang XJ (Eds.). De Gruyter Open. 2015; Chp. 21. doi: 10.13140/RG.2.1.2557.0323.
  • [12] Hardy GH, Wright EM. An Introduction to the Theory of Numbers. Oxford University Press; 2008. ISBN-10: 0199219869, 13: 978-0199219865.
  • [13] Kuo SM, Lee BH, Tian W. Real-Time Digital Signal Processing: Implementations and Applications. John Wiley & Sons; 2013. ISBN: 978-1-118-41432-3.
  • [14] Li C, Dao X, Guo P. Fractional derivatives in complex planes. Nonlinear Analysis: Theory, Methods & Applications. 2009; 71 (5-6): 1857-1869. doi: 10.1016/j.na.2009.01.021.
  • [15] Ortigueira MD. A coherent approach to non-integer order derivatives. Signal Processing. 2006; 86 (10): 2505-2515. doi: 10.1016/j.sigpro.2006.02.002.
  • [16] Ramos RV, Mendes FV. Riemannian quantum circuit. Physics Letters A. 2014; 378 (20): 1346-1349. URL http://dx.doi.org/10.1016/j.physleta.2014.02.008.
  • [17] Riemann GFB. Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (i.e. On the Number of Primes Less Than a Given Magnitude). Monatsberichte der Kniglich Preuischen Akademie der Wissenschaften zu Berlin. 1859.
  • [18] Rokhlin V. A fast algorithm for the discrete Laplace transformation. Journal of Complexity. 1988; 4 (1): 12-32. doi: 10.1016/0885-064X(88)90007-6.
  • [19] Schumayer D, Hutchinson DAW. Physics of the Riemann hypothesis. Reviews of Modern Physics. 2011; 83 (2): 307-330. doi: 10.1103/RevModPhys.83.307.
  • [20] Stirling DSG. Mathematical Analysis and Proof. Woodhead Publishing; 2009. ISBN-10: 1904275400, 13: 978-1904275404.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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