The fractional parts of the Bernoulli numbers
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.
scientific article; zbMATH DE number 3629035