  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.
