Moment inequalities and the Riemann hypothesis
type: Item
instance of: scholarly article
title: Moment inequalities and the Riemann hypothesis
zbMATH Open document ID: 0696.30007
DOI: 10.1007/BF02075457
author: George Csordas
author: Richard S. Varga
published in: Constructive Approximation
publication date: 1988-01-01T00:00:00Z
review text: The Riemann \(\xi\)-function can be expressed as a Fourier transform \[ \xi (x)=8\int^{\infty}_{0}\Phi (t)\cos xt dt=8\sum^{\infty}_{m=0}((- 1)^ m\hat b_ mx^{2m})/(2m)!, \] where the moments are given by \(\hat b_ m=\int^{\infty}_{0}t^{2m}\Phi (t)dt\) (m\(\geq 0)\), and \[ \Phi (t)=\sum^{\infty}_{n=1}(2n^ 4\pi^ 2e^{9t}-3n^ 2\pi e^{5t})\exp (-\pi n^ 2e^{4t}). \] The Riemann hypothesis is equivalent to the statement that \(\xi\) has only real zeros, while a necessary condition for the truth of the Riemann hypothesis is that the Turán inequalities \[ (\hat b_ m)^ 2>((2m-1)/(2m+1))\hat b_{m- 1}\hat b_{m+1} \] hold for \(m\geq 1\), a result now established [see a paper by the authors and \textit{T. S. Norfolk}, Trans. Am. Math. Soc. 296, 521-541 (1986; Zbl 0602.30030)]. In the paper under review, the authors consider a general transform of \(\Phi\) that includes, as a special case, the Fourier transform. They show that the analogues of the Turán inequalities, which also provide necessary conditions for the truth of the Riemann hypothesis, are true. Of primary importance is the result that log \(\Phi\) (\(\sqrt{t})\) is strictly concave on \(0<t<\infty\).
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11N30
zbMATH DE Number: 4140097
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: Turán inequalities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4140097