On the Zudilin-Rivoal theorem (Q2473774)

The main result of this paper is the following result. Theorem: There exists an effective positive constant \(b_0\) such that, for any odd natural number \(b>b_0\) and even natural number \(a>100b\log b\) there is at least one irrational number among the numbers \(\sum_{n=0}^\infty \frac {(-1)^n}{(2n+1)^s}\) where \(s=b+1, b+3, b+5,\cdots ,b+a-1\). The proof makes use of the Mellin transformation.