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arXiv · 0709.2184

Irrationality measure and lower bounds for pi(x)

David Burt·Sam Donow·Steven J. Miller·Matthew Schiffman·Ben Wieland
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Abstract

In this note we show how the irrationality measure of $ζ(s) = π^2/6$ 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 $x/\log x$ as a natural boundary.

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David Burt, Sam Donow, Steven J. Miller, Matthew Schiffman, Ben Wieland. 2014-12-22. Irrationality measure and lower bounds for pi(x). https://arxiv.org/abs/0709.2184

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