primecount VS primesieve

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.

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.