Counting the number of primes

(Click here for a Postscript version of this page.)

It is known since Euclid that there exists an infinite number of primes. A very large number of proofs can be found in the litteratures (this is certainly one of the mathematical result with so many proofs).

The way this infinity behaves is a great mathematical question. More precisely, mathematicians defined the function p(x) by the number of prime numbers less or equal than a given value x :

p(x) = # { p x,    p is prime}.
For example, p(11) = 5 since there are 5 primes less than or equal to 11, which are 2, 3, 5, 7 and 11.

1  The prime number theorem

One of the mathematical breakthrough of the nineteenth century has been the prime number theorem, proven independantly by Hadamard and De La Vallée Poussin in 1896. Roughly speaking, it states that as x becomes big, p(x) is close to x/log(x) (in other words, a random number x has a "probability" 1/log(x) to be prime).

More precisely, the prime number theorem states that

p(x) = Li(x) + O(x exp(-a   _____
log(x)
 
)),        for some positive constant a,
where
Li(x) =
x

2 
dt
log(t)
.
Thus Li(x) is a good approximation of p(x). The error term is in fact related to one of the most famous mathematical unsolved problem, namely the Riemann hypothesis. If the Riemann hypothesis is true, then it was proved in 1901 by von Koch that we have the higher estimate
p(x) = Li(x) + O(x log(x)).
In other words, the Riemann hypothesis says that Li(x) is a very good approximation to p(x).

More information on the prime number theorem will be soon available on this site.

2  Sieves

Studing the distribution of primes can be done by sieving numbers. In the ancient times, Eratosthenes described a sieve which permits to compute the prime number less than a given value x. Considering the list of integers from 2 to x, it consists by eliminating from this list the non trivial multiples of the primes less than x. The non eliminated values are the prime numbers less than x. This sieve is now classical and was extensively used for example as a distributed projet on the net to compute all prime numbers less than 1016 (see Rensselaer's Twin Primes Computing Effort where a sieve has been used to study twin primes).

Sieves can be used to study statistical data on prime numbers, like counting the number of primes, looking for small and large gaps between consecutive primes, counting the number of twin primes (primes p for which p+2 is also a prime), counting primes with a given form, etc.

3  Efficient calculations of p(x)

The cost of classical sieves are close to linear. For example, the use of the classical Sieve of Eratosthenes to compute p(x) can be done in time O(xloglog(x)). More efficient sieve versions are a little quicker, but the asymptotic cost remains close to linear. A two years distributed computation was necessary for example, to compute a certain family of primes less than 1016 with sieving techniques.

Legendre was the first to suggest a method to compute p(x) without locating all the primes less than x. However, his method was not of immediate utility. The first real breakthrough in the p(x) computation was done by Meissel in 1871, who computed p(108) and later p(109) in 1886, which was a value far beyond the known tables of primes at that time. In 1959, D. H. Lehmer simplified Meissel's method to compute p(1010) using an IBM 701. Bohmann in 1972 computed p(1013). Notice that the old calculations p(109), p(1010) and p(1013) turned out to have slight errors, and the later computations were systematically verified by a second computation with different parameters. In 1985, Lagarias, Miller and Odlyzko improved again Meissel method and obtain an algorithm running in time O(x2/3/log(x)) and memory O(x1/3log2(x)) (see [3]). They obtained the value p(4 ×1016). In 1994 Deléglise and Rivat improved the technique once again [2] and saved a factor log(x) in the time cost (but they lost the same factor on the memory cost). They found higher values : p(1017), p(1018) and later, the values p(1019) and p(1020) with a more efficient implementation by Deléglise.

4  Contribute to a p(x) computation

Recently, Xavier Gourdon (one of the two authors web site) made technical improvement to the Meissel, Lagarias, Miller, Odlyzko, Deléglise and Rivat algorithm (see Computation of pi(x) : improvements to the Meissel, Lehmer, Lagarias, Miller, Odlyzko, Deléglise and Rivat method). As a result, some constant factors have been saved and memory usage is lower (no precise asymptotic study has been made until now). The improvement also permits to distribute a large p(x) computation on several machines with a very low memory exchange (Lagarias, Miller and Odlyzko already considered the problem of distributing the computation in [3] but subsequent memory exchange was needed between the machines), provided a precomputation of cost O(x1/2) is made (a description of the improvements will soon be available on this site). The implementation is a program which permitted to obtain a new p(x) record computation in 2000. The value of

pi(1021) = 21 127 269 486 018 731 928
was obtained in 20 days (time equivalent on a Pentium II 350) and 115 Mo of memory was needed. Progress have been in the implementation, permitting to distribute a large p(x) computation on the net. You can contribute to this project (click here), and why not, be a contributor of the first known value x for which p(x) > Li(x) (see below).

5  Discussion of the approximation of p(x) by Li(x)

The prime number theorem states that p(x) is close to the function

Li(x) =
x

2 
dt
log(t)
.
In fact, the Riemann hypothesis is equivalent to the statement that the Li(x) approximation is very good, and more precisely that
p(x) = Li(x) + O(x1/2 log(x)).

The known values of p(x) today agree with this estimate (see the table below), and even agree with a more tight estimate. It is in fact not so easy to have an idea of what should be exactly the error term. At the end of the 1930's, Cramer proposed a model [4] to construct conjectures on distribution of prime numbers. The Cramer model suggest that the prime numbers behave like a random sequence with the same growth constraint (in other words, each number n have a probability 1/log(n) to be prime in this model). According to this model, the number of primes almost should almost surely (with probability 1) satisfy the equality

p(x) = Li(x) + q(x)   
 


2x loglog(x)
log(x)
 
,    with     -1 q(x) 1.
This model suggests to study more precisely the function q(x), defined by
q(x) = p(x)-Li(x)
(2x loglog x/ log x)1/2
.
Riemann hypothesis is then equivalent to the statement that q(x) = O(log(x)3/2/(loglog(x))1/2).

A result of Littewood show that the function q(x) satisfies

|q(x)| > c logloglog(x)
(log(x) loglog(x))1/2
    for a positive constant c
for a sequence of values of x tending to infinity, meaning that q(x) is not too small.

Oscillations

Riemann and Gauss believed that Li(x) > p(x) (which is equivalent to q(x) < 0) for every x > 3, and all the computed values of p(x) today satisfy this inequality. However it has been proved by Littlewood in 1914 that q(x) changes its sign infinitely often. In 1933, S. Skewes established that the first change of sign occurs before the impressive number 10k with k = 101034. This bound has been improved in 1966 by R.S. Lehman who gave the bound 101166. Later in 1985, H. Te Riele proved that the first change of sign occurs before x = 6.69 ×10370. The best bound known so far has been obtained in 2000 by Carter Bays and Richard H. Hudson [1] who proved that the first change of sign occurs before x = 1.39822 ×10316. Of course, such a large value is not reachable today, but the function q(x) have large variations in the last table values, suggesting that a change of sign could occur not too far from this zone. The p(x) project could afford the first known value x for which p(x) > Li(x) ! Finding the first change of sign if even more difficult !

A graphic view of the function Li(x)-p(x)

Using a program written by Xavier Gourdon, Patrick Demichel computed a large number of values of Li(x)-p(x) for x between x = 1015 and x = 7×1015 every 1012. As a result, he obtained a graphic view of this function in this range. As expected, the resulting plotted function is very chaotic.

6  A table of values of p(x)

This table of values of p(x) contains the difference with the function Li(x) (third column), the value of the "normalized" Cramer function

q(x) = p(x)-Li(x)
(2x loglog x/ log x)1/2
.
(see the discussion about this function in the above section), and information about the first calculation (Corrected means that the first computation was wrong and that the value has been corrected).

Remember that the Cramer model implies the conjecture -1 q(x) 1, which is a stronger conjecture than the bound obtained from the Riemann Hypothesis, and that a change of sign of q(x) has not been found yet (see the section Oscillations above).

Note : the following  page contains the description of an interesting algorithm to compute the parity of p(x) in time O(x log(x)), together with a table of the parity of p(x) for some values of x (this page contains errors in the parity of p(x) for some values of x 5×1019; an implementation of the corresponding algorithm was made by Xavier Gourdon and the parities found confirm the values of our table).

A more complete set of values of p(x) is available in our pi(x) table page.

x p(x) Li(x)-p(x) q(x) Comments
106 78,498 129 -0.209
107 664,579 336 -0.1809
108 5,761,455 753 -0.1339 Meissel 1871
109 50,847,534 1,700 -0.0994 Meissel 1886 (Corrected)
1010 455,052,511 3,103 -0.0594 Lehmer 1959 (Corrected)
1011 4,118,054,813 11,587 -0.0725
1012 37,607,912,018 38,262 -0.0781
1013 346,065,536,839 108,970 -0.0723 Bohmann 1972 (Corrected)
1014 3,204,941,750,802 314,889 -0.0678 Lagarias Miller Odlyzko 1985
1015 29,844,570,422,669 1,052,618 -0.0735 LMO 1985
1016 279,238,341,033,925 3,214,631 -0.0726 LMO 1985
1017 2,623,557,157,654,233 7,956,588 -0.0581 Deléglise Rivat 1994
1018 24,739,954,287,740,860 21,949,554 -0.05177 Deléglise Rivat 1994
1019 234,057,667,276,344,607 99,877,774 -0.0760 Deléglise 1996
1020 2,220,819,602,560,918,840 222,744,643 -0.0546 Deléglise 1996
2*1020 4,374,267,703,076,959,271 472,270,046 -0.0823 X. Gourdon 2000 Nov
1021 21,127,269,486,018,731,928 597,394,253 -0.0471 X. Gourdon 2000 Nov
2*1021 41,644,391,885,053,857,293 1,454,564,714 -0.0816 pi(x) project, 2000 Dec
4*1021 82,103,246,362,658,124,007 1,200,472,717 -0.0479 pi(x) project, 2000 Dec
1022 201,467,286,689,315,906,290 1,932,355,207 -0.0491 pi(x) project, 2000 Dec
1.5*1022 299,751,248,358,699,805,270 2,848,114,312 -0.0592 P. Demichel, X. Gourdon, 2001 Feb
2*1022 397,382,840,070,993,192,736 2,732,289,619 -0.0493 pi(x) project, 2001 Feb
4*1022 783,964,159,847,056,303,858 (*) 5,101,648,384 -0.0655 pi(x) project, 2001 Mar (Current world record)

(*) Thanks to Leonid Durman who detected that I had written the value of li(4*1022) instead of p(4*1022). The mistake have been corrected.

7  Related links

References

[1]
Carter Bays and Richard H. Hudson, A new bound for the smallest x with pi(x)>Li(x), Math. Comp. 69 (2000), 1285-1296.
[2]
M. Deléglise and J. Rivat, Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comp., 65 (1996) 235-245 (see here for an abstract of the paper)
[3]
J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): the Meissel-Lehmer method, Math. Comp., 44 (1985) 537-560.
[4]
G. Tenenbaum et M. Mendès France, Les nombres premiers, Collection que sais-je ?, Presses universitaires de France, 1997.


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