Unfortunately (or not), this is still an open problem. Actually, problems involving product of $\pi$ by irrational numbers seem to be out of reach. A typical one is due to Nesterenko involving algebraic independence of $\pi$ and $e^{\pi}$. In this case, Apery or Beukers methods do not work (at least in my mind).

Unfortunately (or not), this is still an open problem. Actually, problems involving product of $\pi$ by transcendental numbers seem to be out of reach. A typical one is due to Nesterenko involving algebraic independence of $\pi$ and $e^{\pi}$. In this case, Apery or Beukers methods do not work (at least in my mind).

Unfortunately (or not), this is still an open problem. Actually, problems involving product of \pi by irrational numbers seems to be out of reach. A typical one is due to Nesterenko involving algebraic independence of \pi and e^{\pi}. In this case, Apery or Beukers methods does not work (at least in my mind).

Unfortunately (or not), this is still an open problem. Actually, problems involving product of $\pi$ by irrational numbers seem to be out of reach. A typical one is due to Nesterenko involving algebraic independence of $\pi$ and $e^{\pi}$. In this case, Apery or Beukers methods do not work (at least in my mind).