The combinatorial algorithm for computing $\pi(x)$

TL;DR

This paper presents improved combinatorial algorithms for calculating the prime counting function π(x), reducing memory usage and enabling large-scale computations up to 10^26 and 2^86, with parallelization enhancements.

Contribution

It introduces a memory-efficient, parallelizable combinatorial method for computing π(x) with proven complexity bounds, enabling large-scale prime counting.

Findings

Memory usage reduced by factor of log x

Successfully computed π(10^n) for n up to 26

Computed π(2^m) for m up to 86

Abstract

This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and distributed-memory parallelism have been incorporated. The resulting method computes $\pi(x)$ with complexity $O(x^{2/3}\mathrm{log}^{-2}x)$ in time and $O(x^{1/3}\mathrm{log}^{2}x)$ in space. The algorithm has been implemented and used to compute $\pi(10^n)$ for $1 \leq n \leq 26$ and $\pi(2^m)$ for $1\leq m \leq 86$. The mathematics presented here is consistent with and builds on that of previous authors.