Irrationality measure and lower bounds for pi(x)
Abstract
In this note we show how the irrationality measure of can be used to obtain explicit lower bounds for . We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for 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 as a natural boundary.
Keywords
Cite
@article{arxiv.0709.2184,
title = {Irrationality measure and lower bounds for pi(x)},
author = {David Burt and Sam Donow and Steven J. Miller and Matthew Schiffman and Ben Wieland},
journal= {arXiv preprint arXiv:0709.2184},
year = {2014}
}
Comments
Version 4.0, 7 ps. E. Kowalski and T. Rivoal point out that the irrationality bound on zeta(2) uses the PNT to bound lcm(1,...,n). We explore this connection in greater detail, and modify the irrationality bound proofs. We can get arbitrarily close to x /log x infinitely often, and thus see that the true value of pi(x) is a natural boundary for these techniques