Abstract
It is well known that the Riemann zeta-function $\zeta(s)$ 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 $s\geq2$ , and for the special values $s=2, 3$ , we give two exact identities for the integer part of the reciprocal sums of the Riemann zeta-function. For general integer $s\geq4$ , we also propose an interesting open problem.
Highlights
Let complex number s = σ + it, if σ >, the famous Riemann zeta-function ζ (s) is defined by the Dirichlet series∞ ζ (s) = ns, n=and it is analytic everywhere except for a simple pole at s = with residue .As regards the various properties of ζ (s), many mathematicians have studied them and obtained abundant research results
Many research results as regards the Riemann zeta-function basically can be summarized in three aspects: (A) the estimation of the order for the Riemann zetafunction; (B) the mean value theorem for the Riemann zeta-function; (C) the zeros density estimation for the Riemann zeta-function
With regard to a most important problem related to the zeros density estimation of the Riemann zeta-function one has the most famous Riemann hypothesis
Summary
With regard to a most important problem related to the zeros density estimation of the Riemann zeta-function one has the most famous Riemann hypothesis. This paper is inspired by [ , ], and [ ], we will study the properties of the Riemann zeta-function from another angle. Ohtsuka and Nakamura [ ] first studied the properties of the reciprocal sums of Fn, and they proved two identities: Fn– Fn – , FnFn– , if n ≥ is even; if n ≥ is odd, where the function [x] denotes the greatest integer ≤ x.
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