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This question is related to Paper of Wadim Zudilin and Tanguy Rivoal "A note on odd zeta values".

Some users are saying that I should not post question related to papers in MO but site mentions questions from graduate text books and papers could be asked. Please give some hint and a question easy to you might not necessarily be easy to me .

Adding images

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I am unable to prove in last line of 4 th image how $(\phi_{n})^{-3} {d_n}^{A+2} q_{j,n}$ is an integer for odd j $\geq$ 5 but first part for $j=0$ I could prove.

I tried 2nd part 3 times but could not prove it. Can someone please help.

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    $\begingroup$ You have a mathematical question, but in my humble opinion it would be much better if you can phrase it differently from just extracting and copy-pasting from the article. Making this effort may also help you understand which theory you need. This probably belongs to MSE (unless you have a specific reason to believe there is some argument missing or some explanation needed (at research level), but you don't mention such a reason here). $\endgroup$ Feb 2, 2020 at 18:19
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    $\begingroup$ I see. General advice I can give you is that when you face a question you have no idea how to solve, try to reformulate it in your own words and notation, detaching it from the context, and transforming it into a natural (possibly more general) statement with no or little notations from the original article. If you still cannot solve it, this will make a fine question for MSE, or MO if it happens to be more difficult than expected. Of course, other people can tell you the solution of the first question, but you will learn and benefit much less than with the "reformatting" process. $\endgroup$ Feb 2, 2020 at 20:19
  • $\begingroup$ Of course I also understand things are not so simple when you don't have easy access to references from your location. $\endgroup$ Feb 2, 2020 at 20:20
  • $\begingroup$ This question already had an accepted answer. Why have you bumped it with an edit? $\endgroup$
    – Yemon Choi
    Jan 6 at 0:42

1 Answer 1

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Expression (6) from the paper is

$$ q_{j,n} = \sum_{m=0}^n (j-2)(j-1)p_{j-2,m} $$

for odd $j\geq 5$. It follows that

$$ \Phi_n^{-3}d_n^{A+2}q_{j,n} = \sum_{m=0}^n (j-2)(j-1)\Phi_n^{-3}d_n^{A+2}p_{j-2,m} \in \mathbb{Z}.$$

The argument before the part you quote shows that

$$ \Phi_n^{-3}d_n^{A-j}p_{j,m} \in \mathbb{Z}$$

for any $1\leq j\leq A$ and any $0\leq m\leq n$. Since $d_n=\mathrm{lcm}(1,2,\ldots,n)$ is an integer we know that

$$(j-2)(j-1)\Phi_n^{-3}d_n^{A+2}p_{j-2,m}\in\mathbb{Z}.$$

As the sum of integers $\Phi_n^{-3}d_n^{A+2}q_{j,n}$ is itself an integer. Although not stated at this point in the proof, this deduction is only valid when $1\leq j-2\leq A$, but the range of $j$ we are looking at for $q_{j,n}$ is odd $j$ with $5\leq j\leq A+1$, so this is valid where needed.

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