primesieve VS primecount

This is a fascinating Q&A where user GordonBGood analyzes the performance of the Sieve of Atkin and compares it to that of Eratosthenes with a view to practical implementations.

The fast prime generator project primesieve is also relevant: https://github.com/kimwalisch/primesieve

I've come across this excellent prime sieve on GitHub, and I just want to find out how it generally works. Yes, it's written in C but I plan to make a Python version that uses some of its methods to make a fairly quick prime sieve. However, I'm really not sure how it has implemented wheel factorisation, and no matter how hard I look online, I can't find a good execution of it that works with its segmented approach. Does anyone have any idea how the wheel factorisation is implemented? To my understanding it's a modulus array that tells you which numbers modulo n are definitely not prime leaving you with the candidate primes to check, but I'm not sure how you would implement this inside a segment so that you only check the candidate primes. In the prime sieve on GitHub it somehow finds the next multiple of the prime using its lookup tables, which I cannot decipher.

Implementation: https://github.com/kimwalisch/primesieve/wiki/Segmented-sieve-of-Eratosthenes

Here another version that still run fast (between 15 and 20ms). A better implementation of the sieve of Eratosthenes is primesieve. You could also use the sieve of Atkin.

Well, that's fine and all, but factually, its very slow: - https://github.com/kimwalisch/primesieve achieve over a billion prime in the same amount of time

The list of primes at primes.utm.edu contains just 50 million primes. A program like Primesieve by Kim Walisch can compute them in a fraction of a second.

There are approximately 193 million primes under 232 which is the square root of 264, you quickly generate a list of all primes using https://github.com/kimwalisch/primesieve - and then do trial division in paralelle on your input set using rayon.

The MathWorld page on the prime counting function says that we learned in 2015 that there are 16,352,460,426,841,680,446,427,399 primes less than 1027; the github for the software used to compute that number says the program will work on any integer up to 1031. Expect it to take a loooong time to run on large numbers.