Counting Primes Rationally And Irrationally

Abstract

The recent technique for estimating lower bounds of the prime counting function $π(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu (\zeta (s))\ge 2$ 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 $\mu (\zeta (s))\ge 1$ for rational ratios of zeta functions.

Eight Pages. Keywords: Distribution of prime; Prime counting function

References in corpus (1)

• Irrationality measure and lower bounds for pi(x)