Infinite Series Concerning Tails of Riemann Zeta Values

Round 1

Reviewer 1 Report

You are missing a lot of steps to follow the proof of each result. I recommend you insert more detailed proofs. I believe most of the results are novel, and some of the particular values are unnecessary, but adding them is acceptable.

Author Response

Thanks to Reviewer 1 for the positive statements.

Apart from the introduction section, each section starts with a class of elementary
symmetric functions that are characterized by a lemma with a detailed proof. The remaining part
of the same section is divided into 4 subsections, where each subsection consists of a main theo-
rem with a detailed derivation (proof) and a corollary, highlighting a number of interesting specific
identities, obtained by extracting coefficients of monomials in u and v. This last step is realized by
Mathemtica, instead of manual derivation, since it is completely routine, but tedious.

 

Reviewer 2 Report

Dear Editor

AXIOMS MDPI

 

Title:  INFINITE SERIES CONCERNING TAILS OF RIEMANN ZETA VALUES

Authors:  CHUNLI LI AND WENCHANG CHU

Reviewer's report

This paper is devoted to present Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails as coefficients of the bivariate beta function Beta  expanded into Laurent series in  and . Several explicit summation formulae are also deduced as consequences. This is realized by integral representations of zeta tails together with the telescopic approach.

In my opinion, the results are novel and the proofs are correct as well. This paper deserves publication and it will make a significant contribution to the literature after publication. The current version of this paper is well written, and the results are quite valuable and very interesting. Therefore, I am very willing to recommend the acceptance of it for publication in AXIOMS. However, it is worth pointing out that there are some suggestions or typo mistakes which need to be modified and corrected:

 

1)       Page 1, Abstract should be improved.

2)       Page 11, Authors should add a conclusion section.

In addition, please the authors add and insert some recent references in this article in order to help readers .

Dear Editor

AXIOMS MDPI

 

Title:  INFINITE SERIES CONCERNING TAILS OF RIEMANN ZETA VALUES

Authors:  CHUNLI LI AND WENCHANG CHU

Reviewer's report

This paper is devoted to present Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails as coefficients of the bivariate beta function Beta  expanded into Laurent series in  and . Several explicit summation formulae are also deduced as consequences. This is realized by integral representations of zeta tails together with the telescopic approach.

In my opinion, the results are novel and the proofs are correct as well. This paper deserves publication and it will make a significant contribution to the literature after publication. The current version of this paper is well written, and the results are quite valuable and very interesting. Therefore, I am very willing to recommend the acceptance of it for publication in AXIOMS. However, it is worth pointing out that there are some suggestions or typo mistakes which need to be modified and corrected:

 

1)       Page 1, Abstract should be improved.

2)       Page 11, Authors should add a conclusion section.

In addition, please the authors add and insert some recent references in this article in order to help readers .

Author Response

Thanks to Reviewer 2 for the positive comments and recommendation.

We have rewritten “Abstract” and added
a “Conclusion section” on page 20. Minor errors and typos are corrected and highlighted by
“Trackchanges”. In addition, we updated references by adding 5 new ones [9–13].

Reviewer 3 Report

In this paper, the authors expressed  infinite series involving Riemann’s zeta and Dirichlet’s lambda tails as coefficients of the bivariate beta function Beta.u; v/, expanded into Laurent series in u and v and some explicit summation formulae will be obtain as consequences after realized by integral representations of zeta tails together with the telescopic approach. The paper is correct and generally correctly written and would help the researchers to get a thorough hold on its topic and its applications. If the authors wish, they can add the following article to the reference sections in order to give different ideas to the readers.

M. Aiyub,, A. Esi and N. Subramanian, Poisson Fibonacci binomial matrix on rough statistical convergence on triple sequences and its rate, Journal of Intelligent & Fuzzy Systems 36 (2019) 3439–3445. 

Author Response

Thanks to Reviewer 3 for the detailed analysis and positive assess-
ments.

Round 2

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

Thanks to Reviewer for the positive statements and the detailed
comments, that are faithfully applied during this round of revision.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

I think the file needs much more improvement. The commands the author inserted do not work with Wolfram Mathematica. I cant verify the identities are correct. After the second revision I don´t think the author takes seriously the review.

Author Response

see uploaded revision notes

Author Response File: Author Response.pdf