Irrationality measure and lower bounds for pi(x)

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In this note we show how the irrationality measure of ζ(s) = π²/6 can be used to obtain explicit lower bounds for π(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 π(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/ x as a natural boundary.