Some identities related to Riemann zeta-function

Abstract It is well known that the Riemann zeta-function ζ ( s ) $\zeta(s)$ plays a very important role in the study of analytic number theory. In this paper, we use the elementary method and some new inequalities to study the computational problem of one kind of reciprocal sums related to the Riemann zeta-function at the integer point s ≥ 2 $s\geq2$ , and for the special values s = 2 , 3 $s=2, 3$ , we give two exact identities for the integer part of the reciprocal sums of the Riemann zeta-function. For general integer s ≥ 4 $s\geq4$ , we also propose an interesting open problem