The fractional parts of the Bernoulli numbers
type: Item
instance of: scholarly article
title: The fractional parts of the Bernoulli numbers
zbMATH Open document ID: 0405.10011
author: Samuel S. jun. Wagstaff
author: P. Erdős
published in: Illinois Journal of Mathematics
publication date: 1980-01-01T00:00:00Z
review text: It is proved that the fractional parts of the Bernoulli numbers \(B_{2 k}\) are dense in the interval \((0,1)\). Furthermore, for every positive integer \(k\), the set of all \(m\) for which \(B_{2 m}\) has the same fractional part as \(B_{2 k}\) as positive asymptotic density. The second statement is proved via this result on divisibility by \(p-1\): For each \(\varepsilon> 0\), there is a \(T=T(\varepsilon)\) so that if \(x> T\), then the number of \(m\leq x\) which have a divisor \(p-1> T\), with \(p\) prime is less than \(\varepsilon x\). The paper concludes with several related open questions.
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 3629035
zbMATH Keywords: positive asymptotic density
zbMATH Keywords: divisibility
zbMATH Keywords: Bernoulli number
zbMATH Keywords: fractional part
zbMATH Keywords: shifted prime
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3629035