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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.

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  • $\begingroup$ I think you need to give me the bounty separately. Thank you in advance! $\endgroup$
    – GH from MO
    Commented May 4, 2020 at 18:28
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    $\begingroup$ @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
    Commented May 4, 2020 at 18:44
  • $\begingroup$ Thanks for the bounty (twice)! $\endgroup$
    – GH from MO
    Commented May 5, 2020 at 22:41

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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, $$\tilde r_n=\sum_{d\in\mathcal{D}}w_d\hat r_{n,d}=\sum_{d\in\mathcal{D}}w_d(d+o(1))r_{n,1}=\left(\sum_{d\in\mathcal{D}}w_d d+o(1)\right)r_{n,1}.$$

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    $\begingroup$ @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
    Commented May 4, 2020 at 18:05

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