I am a masters student and I am interested in number theory. Due to lockdown I have a lot of time and I thought of reading a research paper in Number theory which is " Many Odd zeta values are irrational by " Stephane Fischler, Johannes Sprang and Wadim Zudilin.

I have a question on page 8 just after the (3.6)

My question is -> how authors deduced that $c_k,j \leq (2D)^{3Dn} ( n! / (k)^{ n+1} )^{s+1-3D}$ , for n large.

Adding image->

$c_{k,j}$" />

I am not able to understand how authors came to this conclusion. I think probably using $c_k, j$ 's definition one could get it. I can divide and multiply by $(2)^{3Dn} $ and use that s+1> 3D and that there are n+1 terms in denominator raised to exponent s+1 . These things indicate me that definition of $c_{j, n} $ would be used but I am not getting exact given inequality.

Can someone please tell how it will be derived.

Edit 1-> I have another question , Unfortunately I am unable to derive the inequality given after (3.6) which begins with line using hypothesis (3.1) and Stirling formula . Hypothesis is s/D log D is sufficiently large.

In this I tried by using the relation that k is in denominator so all k $\geq$ $[ ((A(\epsilon) -(\epsilon)) n]$ = x (let) then in denominator I have 1/x* + 1/(x+1)* +..., where * denote [ ( n+1) ( s+1-3D) ] but I have no idea how can I use this sum( how to simplify it and in what way) to get the inequality proved and I don't know how Stirling Approximation formula would be used.

Please help.