Some new upper bounds on the generation of prime numbers

Abstract

Given an integer N, what is the computational complexity of finding all the primes less than N? A modified sieve of Eratosthenes using doubly linked lists yields an algorithm of O A (N) arithmetic complexity. This upper bound is shown to be equivalent to the theoretical lower bound for sieve methods without preprocessing. Use of preprocessing techniques involving space-time and additive-multiplicative tradeoffs reduces this upper bound to O A (N/log logN) and the bit complexity to O B (N logN log log logN). A storage requirement is described using O B (N logN/log logN) bits as well.