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The combinatorial algorithm for computing $π(x)$ (1503.01839v2)
Published 6 Mar 2015 in math.NT and cs.DS
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 $\pi(10n)$ for $1 \leq n \leq 26$ and $\pi(2m)$ for $1\leq m \leq 86$. The mathematics presented here is consistent with and builds on that of previous authors.