An improved sieve of Eratosthenes

TL;DR

This paper introduces an improved sieve of Eratosthenes that reduces space complexity to O(N^{1/3} (log N)^{2/3}) and maintains efficient time, enabling faster prime enumeration and factorization for large intervals.

Contribution

It presents a novel version of the sieve of Eratosthenes with significantly reduced space requirements and applicability to subintervals, extending its utility beyond traditional methods.

Findings

Reduced space complexity to O(N^{1/3} (log N)^{2/3})

Achieves near-linear time for subintervals of size Omega(N^{1/3} (log N)^{2/3})

Enables integer factorization using the sieve method

Abstract

We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and time O(N log N). These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and time at least linear on N. We can also apply our sieve to any subinterval of [1,N] of length Omega(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length Omega(sqrt{N}). Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ties to Voronoi's work on the Dirichlet divisor problem. The advantage of the method here resides in the fact that, because the method we will give is based on the sieve of Eratosthenes, we will also be able to use it to factor integers, and not just to produce lists of consecutive primes.