The integral of the Riemann \(\xi\)-function
type: Item
instance of: scholarly article
zbMATH Open document ID: 1270.11085
author: Jeffrey C. Lagarias
author: David Montague
publication date: 2012-09-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.4348
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 30D10
zbMATH DE Number: 6079048
zbMATH Keywords: Riemann \(\xi\)-function
zbMATH Keywords: de Bruijn-Newman constant
MaRDI profile type: Publication
arXiv ID: 1106.4348
arXiv classification: math.NT
arXiv classification: math.CV
title: The integral of the Riemann \(\xi\)-function
published in: Commentarii Mathematici Universitatis Sancti Pauli
review text: The Riemann \(\xi\)-function is the entire function defined by the formula NEWLINE\[NEWLINE\xi(s) =\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).NEWLINE\]NEWLINE The \(\xi\)-function satisfies the functional equation \(\xi(s)=\xi(1-s)\), and its zeros are exactly the non-trivial zeros of the Riemann zeta function \(\zeta (s)\). The rescaled function \(\Xi(z) :=\xi(1/2+iz)\) has the Fourier integral representation NEWLINE\[NEWLINE \Xi(z):=2\int_{0}^{\infty}\Phi(u)\cos(zu)\,du,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Phi (u):=\sum_{n=1}^{\infty} ( 4\pi^2 n^{4} e^{9u/2} - 6\pi n^{2} e^{5u/2})\exp \left(-\pi n^{2} e^{2u}\right), \quad 0\leq u< \infty. NEWLINE\]NEWLINE In this interesting paper under review, the authors study the integral of the Riemann \(\xi\)-function defined by \(\xi^{(-1)} (s)=\int_{1/2}^s\xi(w)\,dw\), \(s=1/2+iz\). They investigate a one-parameter family of functions given by Fourier integrals which satisfy a functional equation. Members of this family are shown to have only finitely many zeros on the critical line, with \(\xi^{(-1)} (s)\) having exactly one zero on the critical line, \(s = 1/2\). The authors prove that the zeros of \(\xi^{(-1)} (s)\) lie arbitrarily far away from the critical line. The rescaled function \(\Xi^{-1}(z)=-i\xi^{(-1)} (1/2+iz)\) has the Fourier integral representation NEWLINE\[NEWLINE \Xi^{-1}(z):=2\int_{0}^{\infty}\Phi(u) \left(\frac{\sin zu}{u}\right)\, du. NEWLINE\]NEWLINE The authors study the locations of zeros of this function and other entire functions related to \(\xi^{(-1)} (s)\). In addition, they introduce an analogue of the de Bruijn-Newman constant for this family, and prove that it is infinite.NEWLINENEWLINEIt is worth quoting here the authors's perspicuous statement about the purpose of their research. ``To add perspective to the results above, we study the effect of the inverse operation of integration applied to the Riemann \(\xi\)-function on the zeros of the resulting function. Since differentiation …
reviewed by: George Csordas
description: scientific article; zbMATH DE number 6079048