Irrationality Exponents For Even Zeta Constants

Abstract

Let $k\geq 1$ be a small fixed integer. The rational approximations $\left |p/q-\pi^{k} \right |>1/q^{\mu(\pi^k)}$ of the irrational number $\pi^{k}$ are bounded away from zero. A general result for the irrationality exponent $\mu(\pi^k)$ will be proved here. The irrationality exponents for the even parameters $2k$ correspond to those for the even zeta constants $\zeta(2k)$. The specific results and numerical data for a few cases $k=2$ and $k=3$ are also presented and explained.