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 , one may define the following functions: Firstly,

this function measures the set (closed interval) when one has sieved out multiples of the first primes (including these primes themselves); that is, is the sequence of prime numbers in increasing order.

We may further split that up with the functions

these measure the set but considers only the numbers that have exactly prime factors (this is well-defined by the fundamental theorem of arithmetic). With these, we have

the sum on the right is finite, since numbers have, for instance, prime factors.

The identities

prove that one may compute by computing and , . And this is what the Meissel–Lehmer algorithm does.

Formulae for the Pk(x, a)

For , we get the following formula for :

For , there is a similar expansion.[2]

Expanding 𝜑(x, a)

Using the formula

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 and for certain values of and . 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 in time and space for any .[1] Upon setting , the tree of has leaf nodes.[1]

References

  1. Lagarias, Jeffrey; Miller, Victor; Odlyzko, Andrew (April 11, 1985). "Computing : The Meissel–Lehmer method" (PDF). Mathematics of Computation. 44 (170): 537–560. doi:10.1090/S0025-5718-1985-0777285-5. Retrieved September 13, 2016.
  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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