Complexity
The sieve of Eratosthenes is generally considered the easiest sieve to implement, but it is not the fastest. It can find all the primes up to N in time O(N), while the sieve of Atkin and most wheel sieves run in sublinear time O(N/log log N). The sieve of Atkin takes space N1/2+o(1); segmented Eratosthenes' sieve takes slightly less space than O(N1/2). Sorenson shows an improvement to the wheel sieve that takes even less space at O(N/((log N)Llog log N) for any L > 1.
Read more about this topic: Generating Primes
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