Irrationality measure and lower bounds for pi(x)

TL;DR

This paper explores how the irrationality measure of the Riemann zeta function at 2 can be used to derive explicit lower bounds for the prime counting function pi(x), providing elementary proofs and a natural boundary of order x/log x.

Contribution

It introduces a more elementary approach to connect the irrationality measure of zeta(2) with lower bounds for pi(x), achieving a natural boundary of order x/log x.

Findings

Derived explicit lower bounds for pi(x) from irrationality measures.

Provided elementary proofs for bounds related to pi(x).

Established a natural boundary of order x/log x for pi(x).

Abstract

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.