Meissel-Lehmer method

From Polymath1Wiki
Jump to: navigation, search

The Meissel-Lehmer method is a combinatorial method for computing [math]\pi(x)[/math] in time/space [math]x^{2/3+o(1)}[/math]. It is analysed at LMO???

Define an [math]x^{1/3}[/math]-almost prime to be an integer less than x not divisible by any prime less than [math]x^{1/3}[/math]; such a number is either a prime, or a product of two primes which are between [math]x^{1/3}[/math] and [math]x^{2/3}[/math]. Each number of the latter form can be written in two ways as the product of a prime p between [math]x^{1/3}[/math] and [math]x^{2/3}[/math], and a prime between [math]x^{1/3}[/math] and [math]x/p[/math], except for the squares of primes between [math]x^{1/3}[/math] and [math]x^{1/2}[/math] which only have one such representation. The number [math]\pi(x,x^{1/3})[/math] of almost primes less than x is thus

[math] \pi(x) - \pi(x^{1/3}) + \frac{1}{2} \sum_{x^{1/3} \leq p \leq x^{2/3}} [\pi( x/p ) - \pi(x^{1/3})] + \frac{1}{2} (\pi(x^{1/2})-\pi(x^{1/3}))[/math] (1)

(ignoring some floor and ceiling functions).

Using the sieve of Erathosthenes, one can compute [math]\pi(y)[/math] for all [math]y \leq x^{2/3}[/math] in time/space [math]x^{2/3+o(1)}[/math], so every expression in (1) is computable in this time except for [math]\pi(x)[/math]. So it will suffice to compute the number of [math]x^{1/3}[/math]-almost primes less than x in time [math]x^{2/3+o(1)}[/math].

If we let [math]\pi(x,a)[/math] denote the number of a-almost primes less than x (i.e. the number of integers less than x not divisible by any integer between 2 and a), we have the recurrence

[math]\pi(x,a) = \pi(x,a-1) - \pi(x/a, a)[/math] (2)

which reflects the basic fact that the a-1-almost primes are the union of the a-almost primes, and the a-1-almost primes multiplied by a. On the other hand, by factoring all the numbers up to [math]x^{1/3}[/math] by the sieve of Eratosthenes, we can store [math]\pi(y,a)[/math] for all [math]a, y \leq x^{1/3}[/math], to be retrievable in O(1) time. Meanwhile, by recursively expanding out (2) until the x parameter dips below [math]x^{1/3}[/math], one can express [math]\pi(x,x^{1/3})[/math] as an alternating sum of various [math]\pi(y,b)[/math] with [math]y,b \leq x^{1/3}[/math], with each [math]\pi(y,b)[/math] occurring at most once; summing using the stored values then gives [math]\pi(x,x^{1/3})[/math] as required.