On integer parts of the reciprocal remainders of some sums

Abstract

We present generalizations of some results on the integer parts of the reciprocal remainders of the zeta function zeta (s) with s=2 and s=3, and a very short and elegant proof of a recent result on the integer parts of the reciprocal remainders of the series zeta (3). We also give some historical and theoretical remarks to problems of this type, conduct some analyses, and make some connections with the theory of linear difference equations with constant coefficients.