Remarks on Weil's quadratic functional in the theory of prime numbers. I
type: Item
instance of: scholarly article
title: Remarks on Weil's quadratic functional in the theory of prime numbers. I
zbMATH Open document ID: 1008.11034
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2002-12-17T00:00:00Z
review text: Weil's explicit formula, in the case of the Riemann zeta-function \(\zeta(s)\), can be stated as follows. Let \(f\in C_0^{\infty}((0,\infty))\) be a smooth complex-valued function with compact support in \((0,\infty)\), let \(\widetilde{f}(s)=\int_0^{\infty} f(x)x^{s-1} dx\) be the Mellin transform of \(f\) and \(f^*(x)=\frac 1{x}f(\frac 1{x})\). Then \[ \sum_{\rho} \widetilde f(\rho) =\int_0^{\infty} f(x) dx + \int_0^{\infty} f^*(x) dx -\sum_{n=1}^{\infty} \Lambda(n)\{f(n)+f^*(n)\} - \] \[ -(\log 4\pi +\gamma)f(1)-\int_1^{\infty} \biggl\{f(x)+f^*(x)- \frac{2}{x}f(1)\biggr\} \frac{x dx}{x^2-1}, \] where the sum on the left hand side is over all complex zeros of \(\zeta(s)\) and \(\gamma\) is Euler's constant. Denote by \(\mathcal T(f)\) the right hand side of the above explicit formula. \(\mathcal T(f)\) is a linear functional and \(\mathcal T(f*\overline{f}^*)\) is called Weil's quadratic functional, where ``\(*\)'' stands for the multiplicative convolution of functions. \textit{A. Weil} [Meddel. Lunds Univ. Mat. Sem. (dedié à M. Riesz), 252-265 (1952; Zbl 0049.03205)] showed that \(\mathcal T(f*\overline{f}^*)\) is positive semidefinite if and only if the Riemann Hypothesis (R.H.) is true. This paper studies Weil's functional from a variational point of view, restricted to two classes of Hilbert spaces, sufficiently wide for testing the validity of the R.H. It is proved that the infimum of \(\mathcal T(f*\overline{f}^*)\) is attained in the unit sphere of the \(L^2\)-space of functions with support in a given interval \([-t,t]\). Furthermore, let \(W_0^{1,2}([M^{-1},M])\) be the space of functions \(f\) with \(f, Df\in L^2((0,\infty))\) and compact support in \([M^{-1},M]\), with norm \[ \|f\|'=\left( \int_{M^{-1}}^M |Df(x)|^2 \right) ^{1/2} dx, \] where \(D\) is the translation invariant differential operator \(D = x(d/dx)\). If the infimum of \(\mathcal T(f*\overline{f}^*)\) in the unit sphere of the space \(W_0^{1,2}([M^{-1},M])\) is negative, then this infimum…
reviewed by: László Tóth
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1846845
zbMATH Keywords: Weil's quadratic functional
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: Riemann Hypothesis
zbMATH Keywords: Explicit Formula
zbMATH Keywords: Fourier transform
zbMATH Keywords: quadratic form
zbMATH Keywords: eigenfunction
MaRDI profile type: Publication
author: Enrico Bombieri
description: scientific article; zbMATH DE number 1846845