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April 6th, 2018, 12:45 PM   #1
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Prime counting. Meissel, Lehmer: is there a general formula?

I am looking for a general formula to count prime numbers on which the Meissel and Lehmer formulas are based:
$$\pi(x)=\phi(x,a)+a-1-\sum\limits_{k=2}^{\lfloor log_2(x) \rfloor}{P_k(x,a)}$$

Wiki - prime counting - Meissel Lehmer

More precisely, I am looking for the detailed description of the $P_k$ for $k>3$.

$P_k(x,a)$ counts the numbers $<=x$ with exactly $k$ prime factors all greater than $p_a$ ($a^{th}$ prime), but in the full general formula, this last condition is not necessary.

The Meissel formula stops at $P_2$ (and still uses some $\phi$/Legendre parts)
Wolfram - Meissel

The Lehmer formula stops at $P_3$ (and still uses some $\phi$/Legendre parts)
Wolfram - Lehmer

I don't find anything about the general formula (using all the $P_k$ terms).
Is there any paper on it?
Why stop at $P_3$? is it a performance issue?

Lehmer vaguely talk about it in his 1959 paper
On the exact number of primes less than a given limit

Deleglise talks about performances here
Prime counting Meissel, Lehmer, ...

Thanks

Last edited by skipjack; April 6th, 2018 at 02:46 PM.
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