Discrete measures and the Riemann hypothesis
type: Item
instance of: scholarly article
title: Discrete measures and the Riemann hypothesis
zbMATH Open document ID: 0822.11061
author: Alberto Verjovsky
published in: Kodai Mathematical Journal
publication date: 1995-10-23T00:00:00Z
review text: The author's main theorem may be restated as follows. Let \({1\over 2}\leq \gamma\leq 1\); the Riemann zeta-function has no zeros in the half-plane \(\text{Re } s>\gamma\) if and only if \(\sum_{n=1}^ \infty \varphi (n) f({n\over x})= m_ 0 (f,x)+ O(x^{\gamma+ \varepsilon})\), \(\varepsilon>0\), for any smooth function \(f: \mathbb{R}_ +\to \mathbb{R}\) with compact support; here \(\varphi\) denotes Euler's function, \(\mathbb{R}_ +\) is the set of positive real numbers, and \(m_ 0 (f,x):= 6\pi^{-2} x^ 2 \int_ 0^ \infty uf(u) du\). The proof makes use of the standard properties of the Mellin transform and the Riemann zeta-function.
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 729253
zbMATH Keywords: Euler's phi-function
zbMATH Keywords: smooth function with compact support
zbMATH Keywords: Mellin transform
reviewed by: B. Z. Moroz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138040054
OpenAlex ID: W2000739120
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DOI: 10.2996/KMJ/1138040054
description: scientific article; zbMATH DE number 729253