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

TL;DR

This paper introduces an enhanced analytic approach for computing the prime counting function $(x)$, successfully calculating $(10^{25})$, advancing the accuracy and efficiency of prime number enumeration.

Contribution

The paper presents a novel improved analytic method for calculating $(x)$, achieving higher precision and computational efficiency.

Findings

Successfully computed $(10^{25})$ using the new method

Demonstrated improved accuracy over previous techniques

Collaborated with experts to validate results

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 $\pi(10^{25})$.