# Understanding a deduction in research paper of Sprang, Fischler and Zudilin ("Many Odd zeta values are irrational")

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:

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.

• I think you need to give me the bounty separately. Thank you in advance! May 4, 2020 at 18:28
• @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. May 4, 2020 at 18:44
• Thanks for the bounty (twice)! May 5, 2020 at 22:41

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