An improved analytic Method for calculating $\pi (x)$

Abstract

We present an improved version of the analytic method for calculating $\pi (x)$, the number of prime numbers not exceeding $x$. We implemented this method in cooperation with J. Franke, T. Kleinjung and A. Jost and calculated the value $\pi (10^{25})$.

arXiv comment21 pages, v2 is a rewrite of v1, title changed from 'A practical analytic method for calculating $\pi (x)$ II' to 'An improved analytic Method for calculating $\pi (x)$