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Let $\pi(x)$ be the number of primes not greater than $x$.

Wikipedia article says that $\pi(10^{23}) = 1,925,320,391,606,803,968,923$.

The question is how to calculate $\pi(x)$ for large $x$ in a reasonable time? What algorithms do exist for that?

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The most efficient prime counting algorithms currently known are all essentially optimizations of the method developed by Meissel in 1870, e.g. see the discussion here http://primes.utm.edu/howmany.shtml

You can use inclusion exclusion principle to get a boost over the Eratosthenes sieve

The Sieve of Atkin is one of the fastest algorithm used to calculate $pi(x)$. The Wikipedia page says that its complexity is O(N/ log log N).

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I found a distributed computation project which was able to calculate $pi(4\times 10^{22})$, maybe it could be useful.

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