An equivalent criteria for irrationality of $\zeta (5)$

Abstract

Defining a Beukers [1] like integral for $\zeta (5)$ as \begin{equation*} I_n:=\int_{(0,1)^5}\frac{(1-x_3)^n(1-x_4)^n P_n(x_1)P_n(x_2)}{1-(1-x_1x_2x_3x_4)x_5} \ dx_1dx_2dx_3dx_4dx_5 \end{equation*} we prove that for each $n\in \mathbb{N}$ \begin{equation*} I_n= \frac{p_nζ(5)+q_nζ(4)+r_nζ(3)+s_n}{d_n^5} \end{equation*} where $p_n,q_n,r_n,s_n$ are integers and $d_n=\text{lcm}(1,2,...,n)$. We prove that the following are equivalent: 1. $q_n\zeta (4)+r_n\zeta (3)-d_n^5{I}_n\notin \mathbb{Z}$ for each natural number $n$. 2. $q_n\zeta (4)+r_n\zeta (3)-d_n^5{I}_n\notin \mathbb{Z}$ for infinitely many natural number $n$. 3. $\zeta (5)$ is irrational.

Not a good article as of now. Needs improvement