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On an Exact Relation between ζ″(2) and the Meijer G -Functions

Instituto Universitario de Matemática Multidisciplinar, Building 8G, 2o Floor, Camino de Vera, Universitat Politècnica de València, 46022 Valencia, Spain
Mathematics 2019, 7(4), 371; https://doi.org/10.3390/math7040371
Received: 25 March 2019 / Revised: 12 April 2019 / Accepted: 15 April 2019 / Published: 24 April 2019
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Abstract

In this paper we consider some integral representations for the evaluation of the coefficients of the Taylor series for the Riemann zeta function about a point in the complex half-plane ( s ) > 1 . Using the standard approach based upon the Euler-MacLaurin summation, we can write these coefficients as Γ ( n + 1 ) plus a relatively smaller contribution, ξ n . The dominant part yields the well-known Riemann’s zeta pole at s = 1 . We discuss some recurrence relations that can be proved from this standard approach in order to evaluate ζ ( 2 ) in terms of the Euler and Glaisher-Kinkelin constants and the Meijer G -functions. View Full-Text
Keywords: Riemann zeta function; Euler-Maclaurin summation; Meijer ?-functions Riemann zeta function; Euler-Maclaurin summation; Meijer ?-functions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Acedo, L. On an Exact Relation between ζ″(2) and the Meijer G -Functions. Mathematics 2019, 7, 371.

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