I am looking for a general forumla to count prime numbers on which the Meissel and Lehmer formula are based:
$$\pi(x)=\phi(x,a)+a-1-\sum\limits_{k=2}^{\lfloor log_{p_{(a+1)}}(x) \rfloor}{P_k(x,a)}$$
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$\leqslant x$ 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
Edit: by "a general formula on which the Meissel and Lehmer formula are based", I meant the one they tend to (with all $P_k$), not the one they started from (Legendre, with no $P_k$). Sorry if it was not clear.