My Math Forum Prime counting. Meissel, Lehmer: is there a general formula?

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 April 6th, 2018, 01:45 PM #1 Newbie   Joined: Sep 2017 From: Belgium Posts: 9 Thanks: 2 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 03:46 PM.

 Tags counting, formula, general, lehmer, meissel, prime

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