# Counting Primes Rationally And Irrationally

## TL;DR

This paper improves lower bounds on the prime counting function using irrationality measures of zeta function values, extending previous results and analyzing rational ratios of zeta functions.

## Contribution

It enhances the lower bound estimate of c(x) to c(x) d; it also extends analysis to rational ratios of zeta functions with lower irrationality measures.

## Key findings

- Lower bound of c(x) improved to c(x) \u2265 b1 \, ext{log} \, x
- Extension of analysis to rational ratios of zeta functions
- Irrationality measures b3(c) d; b3(c) d; b3(c) d; b3(c) d;

## Abstract

The recent technique for estimating lower bounds of the prime counting function $\pi(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s)) \geq 2$ of special values of the zeta function claims that $\pi(x) \gg \log \log x/\log \log \log x$. This note improves the lower bound to $\pi(x) \gg \log x$, and extends the analysis to the irrationality measures $\mu(\zeta(s)) \geq 1$ for rational ratios of zeta functions.