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

• Main Author: Luis Acedo
• Format: Article
• Language: English
• Published: MDPI AG 2019-04-01
• Series: Mathematics
• Subjects: Riemann zeta function; Euler-Maclaurin summation; Meijer ?-functions
• Online Access: https://www.mdpi.com/2227-7390/7/4/371
• ISSN: 2227-7390

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&real;</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Using the standard approach based upon the Euler-MacLaurin summation, we can write these coefficients as <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">&#915;</mi> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> plus a relatively smaller contribution, <inline-formula> <math display="inline"> <semantics> <msub> <mi>&#958;</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula>. The dominant part yields the well-known Riemann&#8217;s zeta pole at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. We discuss some recurrence relations that can be proved from this standard approach in order to evaluate <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>&#950;</mi> <mrow> <mo>&#8243;</mo> </mrow> </msup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> in terms of the Euler and Glaisher-Kinkelin constants and the Meijer <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">G</mi> </semantics> </math> </inline-formula>-functions.