Some identities related to Riemann zeta-function
- Lin Xin^{1}Email author
https://doi.org/10.1186/s13660-016-0980-9
© Xin 2016
Received: 14 December 2015
Accepted: 19 January 2016
Published: 28 January 2016
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.
Keywords
Riemann zeta-function inequality function \([x]\) identity elementary methodMSC
11B83 11M061 Introduction
As regards the various properties of \(\zeta(s)\), many mathematicians have studied them and obtained abundant research results. Some related work can be found in [1–3], and [4]. However, 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 zeta-function; (B) the mean value theorem for the Riemann zeta-function; (C) the zeros density estimation for the Riemann zeta-function. Particularly 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 [5, 6], and [7], we will study the properties of the Riemann zeta-function from another angle.
Some other results related to recursive sequence, recursive polynomial and their promotion forms can also be found in [8–15], here no longer list them one by one.
The main purpose of this paper is to study this problem, and use the elementary method and some new inequalities to give two interesting identities for (1) with \(s=2\) and 3. That is, we shall prove the following two conclusions.
Theorem 1
Theorem 2
Therefore, how to give a precise calculation formula for (1) with \(s=4\) is a very complicated problem. So we propose the following.
Open problem
For integer \(s=4\), does there exist an exact computational formula for (1)?
We hope people who are interested in this problem can study it together with us, and solve this problem finally.
2 Several lemmas
In this section, we shall give some simple lemmas, which are necessary in the proofs of our theorems. First we have the following inequality.
Lemma 1
Proof
Lemma 2
Proof
Lemma 3
Proof
3 Proof of the theorems
Declarations
Acknowledgements
The author would like to thank the referee for very helpful and detailed comments, which have significantly improved the presentation of this paper. This work was supported by the P. S. F. C. (Grant No. 2013JZ001) and N. S. F. C. (Grant No. 11371291).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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