On an Exact Relation between <i>ζ</i>″(2) and the Meijer <inline-formula> <mml:math display="block" id="mm1000"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">G</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>-Functions

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 ℜ ( 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.

Keywords

• Riemann zeta function
• Euler-Maclaurin summation
• Meijer ?-functions