Irrationality measure and lower bounds for pi(x)

In this note we show how the irrationality measure of $\zeta(s) = \pi^2/6$ can be used to obtain explicit lower bounds for $\pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for $\pi(x)$ from these arguments as well. While versions of some of the results here have been done by other authors, our arguments are more elementary and yield a lower bound of order $x/\log x$ as a natural boundary.