I am a master's student interested in number theory. I am reading a research paper in Analytic Number theory which is "Many Odd zeta values are irrational" by Stephane Fischler, Johannes Sprang and Wadim Zudilin.

I have a question about article 5 of this research paper on page 14.

A screenshot of which is posted below:

enter image description here

In the highlighted part of screenshot screenshot I am unable to deduce

$ \hat{r}_{n, d} = ( d+o(1) ) r_{n, 1} $ as $n \rightarrow \infty$, and $\tilde{r}_n = \left(\sum_{d\epsilon D } w_d d +o(1) \right) r_{n,1} $

despite trying a lot.

Any help would be greatly appreciated.

  • $\begingroup$ I think you need to give me the bounty separately. Thank you in advance! $\endgroup$
    – GH from MO
    May 4, 2020 at 18:28
  • 1
    $\begingroup$ @GH from MO yes, you answered 2 questions. By Stackexchange rules bounty can be awarded only after 24 hours of starting it. So, It can't be given to you before 24 hours. But don't worry I will not forget to award you. I am really thankful to you for helping. $\endgroup$
    – Arnold
    May 4, 2020 at 18:44
  • $\begingroup$ Thanks for the bounty (twice)! $\endgroup$
    – GH from MO
    May 5, 2020 at 22:41

1 Answer 1


I am not sure where the difficulty lies here. By (3.2) and the display above the first highlighted item, $$\hat r_{n,d}=\sum_{j=1}^d r_{n,jD/d}=\sum_{j=1}^d (1+o(1))r_{n,1}=(d+o(1))r_{n,1}.$$ Combining this result with the second display above the second highlighted item, $$\tilde r_n=\sum_{d\in\mathcal{D}}w_d\hat r_{n,d}=\sum_{d\in\mathcal{D}}w_d(d+o(1))r_{n,1}=\left(\sum_{d\in\mathcal{D}}w_d d+o(1)\right)r_{n,1}.$$

  • 2
    $\begingroup$ @YannicMuller: Well, Lemma 3 is emphasized below the first highlighted item, and (3.2) is the statement of Lemma 3. I answered your other question as well. Two further remarks. First, please always use a high-level tag like "nt.number-theory". Second, if you are a master's student, then you have an advisor who should (and should be able to) help you out with reading this paper. $\endgroup$
    – GH from MO
    May 4, 2020 at 18:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.