Unable to deduce an inequality in paper on odd zeta values of Fischler, Sprang and Zudilin

Question

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.

!inequalities in which I have question ]1

!defination of $c_{k,j}$]2

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.

Edit -> Unfortunately there is one more question I am having. I am not able to derive the inequality which is just after the line Using (3.1) and Stirling Approximation .

Can someone please tell how it will be derived.

I have tried it many times.

Fischler, Stéphane; Sprang, Johannes; Zudilin, Wadim, Many odd zeta values are irrational, Compos. Math. 155, No. 5, 938-952 (2019). ZBL1430.11097.

Answer 1

1. Let us prove the first inequality. From the definition of $c_{k,j}$, it is clear that $$c_{k,j}\leq D^{3Dn} n!^{s+1-3D} (k+3n+1)^{3Dn+1}k^{-(s+1)(n+1)},$$ hence it suffices to verify that $$(k+3n+1)^{3Dn+1}\leq 2^{3Dn}k^{3D(n+1)}.$$ As explained in the paper, $k$ is much larger than $n$, and $n$ is itself large. Hence $k+3n+1$ is at most $2k$, and it suffices to show that $$(2k)^{3Dn+1}\leq 2^{3Dn}k^{3D(n+1)}.$$ This reduces to $2\leq k^{3D-1}$, which is obvious.

2. Let us prove the second inequality. We shall abbreviate $A(\varepsilon)-\varepsilon$ by $B(\varepsilon)$. By Stirling's approximation, $n!<(n/e)^{n+1}$ for $n$ large, hence it suffices to show that $$2(2D)^{3Dn}\frac{(n/e)^{(n+1)(s+1-3D)}}{(B(\varepsilon)n)^{(n+1)(s+1-3D)-2}}\leq\left(\frac{2D}{eB(\varepsilon)}\right)^{sn/2}.$$ Equivalently, $$2(2D)^{3Dn}\frac{(B(\varepsilon)n)^2}{(eB(\varepsilon))^{(n+1)(s+1-3D)}}\leq\left(\frac{2D}{eB(\varepsilon)}\right)^{sn/2}.$$ For this it suffices that $$2(2D)^{3Dn}\leq(2D)^{sn/2}\tag{1}$$ and $$(eB(\varepsilon))^{sn/2}(B(\varepsilon)n)^2\leq(eB(\varepsilon))^{(n+1)(s+1-3D)}.\tag{2}$$ Both $(1)$ and $(2)$ follow from the fact that $s>6D$ and $n$ is large.