Meissel–Lehmer algorithm

The Meissel–Lehmer algorithm (after Ernst Meissel and Derrick Henry Lehmer) is an algorithm that computes the prime-counting function.^[1]^[2]

Description

The key identity

Given $a\in \mathbb {N}$ , one may define the following functions: Firstly,

\varphi (x,a):=\left|\left([1,x]\cap \mathbb {N} \right)\setminus \bigcup _{j=1}^{a}p_{j}\mathbb {Z} \right|;

this function measures the set $[1,x]\cap \mathbb {N}$ (closed interval) when one has sieved out multiples of the first $a$ primes $p_{1},\ldots ,p_{a}$ (including these primes themselves); that is, $p_{1},p_{2},p_{3},\ldots =2,3,5,\ldots$ is the sequence of prime numbers in increasing order.

We may further split that up with the functions

P_{k}(x,a):=\left|\left[\left([1,x]\cap \mathbb {N} \right)\setminus \bigcup _{j=1}^{a}p_{j}\mathbb {Z} \right]\cap \{n\mid n{\text{ has exactly }}k{\text{ prime factors}}\}\right|,

$k\in \mathbb {N} _{0};$

these measure the set $\left([1,x]\cap \mathbb {N} \right)\setminus \bigcup _{j=1}^{a}p_{j}\mathbb {Z}$ but considers only the numbers that have exactly $k$ prime factors (this is well-defined by the fundamental theorem of arithmetic). With these, we have

\varphi (x,a)=\sum _{k=0}^{\infty }P_{k}(x,a);

the sum on the right is finite, since numbers $\leq x$ have, for instance, $\leq \lfloor x\rfloor$ prime factors.

The identities

\pi (x)=P_{1}(x,a)+a=a+\varphi (x,a)-1-\sum _{k=2}^{\infty }P_{k}(x,a)\qquad {\text{(note }}P_{0}(x,a)=1)

prove that one may compute $\pi (x)$ by computing $\varphi (x,a)$ and $P_{k}(x,a)$ , $k\geq 2$ . And this is what the Meissel–Lehmer algorithm does.

Formulae for the P_k(x, a)

For $k=2$ , we get the following formula for $P_{k}(x,a)$ :

{\begin{aligned}P_{2}(x,a)&=\left|\left\{n\mid n\leq x,n=p_{b}p_{c},a<b\leq c\right\}\right|\\&=\sum _{b=a+1}^{\pi (x^{1/2})}\left|\left\{n~{\big |}~n=p_{b}p_{c},b\leq c\leq \pi \left({\frac {x}{p_{b}}}\right)\right\}\right|\\&=\sum _{b=a+1}^{\pi (x^{1/2})}\left(\pi \left({\frac {x}{p_{b}}}\right)-(b-1)\right)\\&=\sum _{b=a+1}^{\pi (x^{1/2})}\pi \left({\frac {x}{p_{b}}}\right)+{\binom {a}{2}}-{\binom {\pi (x^{1/2})}{2}}.\end{aligned}}

For $k=3$ , there is a similar expansion.^[2]

Expanding ϕ(x, a)

Using the formula

\varphi (x,a)=\varphi (x,a-1)-\varphi \left({\frac {x}{p_{a}}},a-1\right),

$\varphi (x,a)$ may be expanded. Each summand, in turn, may be expanded in the same way, so that a very large alternating sum arises.

Combining the terms

The only thing that remains to be done is evaluating $\varphi (x,a)$ and $P_{k}(x,a),k=1,2,\ldots$ for certain values of $x$ and $a$ . This can be done by direct sieving and using the above formulas.

A modern variant

Jeffrey Lagarias, Victor Miller and Andrew Odlyzko published a realisation of this algorithm which computes $\pi (x)$ in time $O(x^{2/3+\varepsilon })$ and space $O(x^{1/3+\varepsilon })$ for any $\varepsilon >0$ .^[1] Upon setting $a=\pi (x^{2/3})$ , the tree of $\varphi (x,a)$ has $O(x^{2/3})$ leaf nodes.^[1]

References

1 2 3 Lagarias, Jeffrey; Miller, Victor; Odlyzko, Andrew (April 11, 1985). "Computing $\pi (x)$ : The Meissel–Lehmer method" (PDF). Mathematics of Computation. 44 (170): 537–560. doi:10.1090/S0025-5718-1985-0777285-5. Retrieved September 13, 2016.
1 2 Lehmer, Derrick Henry (April 1, 1958). "ON THE EXACT NUMBER OF PRIMES LESS THAN A GIVEN LIMIT". Illinois J. Math. 3 (3): 381–388. Retrieved February 1, 2017.

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[lagarias-miller-odlyzko-1] 1 2 3 Lagarias, Jeffrey; Miller, Victor; Odlyzko, Andrew (April 11, 1985). "Computing $\pi (x)$ : The Meissel–Lehmer method" (PDF). Mathematics of Computation. 44 (170): 537–560. doi:10.1090/S0025-5718-1985-0777285-5. Retrieved September 13, 2016.

[lehmer-2] 1 2 Lehmer, Derrick Henry (April 1, 1958). "ON THE EXACT NUMBER OF PRIMES LESS THAN A GIVEN LIMIT". Illinois J. Math. 3 (3): 381–388. Retrieved February 1, 2017.