Series transformations and integrals involving the Riemann \(\Xi \)-function
type: Item
instance of: scholarly article
title: Series transformations and integrals involving the Riemann \(\Xi \)-function
zbMATH Open document ID: 1206.11104
author: Atul Dixit
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-19T00:00:00Z
review text: Let \(\psi(x):=\frac{\Gamma'(x)}{\Gamma(x)}=-\gamma-\sum_{m=0}^{\infty}\big(\frac{1}{m+x}-\frac{1}{m+1}\big)\) and \(\xi(s):=(s-1)\pi^{-1/2s}\Gamma\big(1+\frac{1}{2}s\big)\zeta(s)\) be the logarithmic derivative of the gamma-function \(\Gamma(x)\) and the Riemann \(\xi\)-function, respectively (here \(\gamma\) is Euler's constant, and \(\zeta(s)\) is the Riemann zeta-function). Then the Riemann \(\Xi\)-function is defined by the formula \(\Xi(t):=\xi\bigl(\tfrac12+it\bigr). \) The paper is devoted to the unification the transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar in the sense that all theses formulas come back from the general formula involving an integral of the \(\Xi\)-function. In the proof of these formulas, the theory of Mellin transforms and the residue theorem are used. Also, the new extensions for the formulas of Koshliakov and Ferrar via their connection with integrals involving the Riemann \(\Xi\)-function are obtained.
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 5710092
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Riemann \(\Xi\)-function
zbMATH Keywords: psi function
zbMATH Keywords: modified Bessel function
zbMATH Keywords: residue theorem
zbMATH Keywords: Mellin transform
zbMATH Keywords: Ramanujan
zbMATH Keywords: Hardy.
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.03.047
OpenAlex ID: W2074946841
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DOI: 10.1016/J.JMAA.2010.03.047
description: scientific article; zbMATH DE number 5710092