At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational

In this article, we prove the irrationality of at least one of the three numbers $ζ(5), ζ(7)$ and $ζ(9)$ is irrational. We find linear forms in $1,ζ(5),ζ(7)$ and $ζ(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $Φ$ involving sp