Prime Number Theorem

Ulam_spiral_primesThe Prime Number Theorem (PNT) describes the distribution of prime numbers. Euclid could prove that there is an infinite number of primes, but their location can only be predicted by statistical means, as an approximation.  As ordinary numbers get larger and larger,  we find fewer and fewer prime numbers scattered among them. This can easily be demonstrated with the “Sieve of Eratosthenes.”  It has resulted in a hunt to find a formula for the location of primes, which is also essential for encrypting messages send across public information highways. Evewn though we cannot predict primes (yet?), we find that the distribution of prime numbers is asymptotic, and follows very precise laws.  In simple terms, the Prime Number Theorem says that, as x approaches infinity, the quotient between x and ln(x)  becomes a better and better estimate of the number of primes at or below x.  That’s not a lot, but it’s as far as we got, and there is still a lot more we can find out about primes.

Here is some more math:

For any positive real number x, we define

\pi(x)={\rm the\ number\ of\ primes\ that\ are\ }\leq x

The prime number theorem then states that


where ln(x) is the natural logarithm of x. This notation means that the limit of the quotient of the two functions π(x) and x/ln(x) as x approaches infinity is 1; it does not mean that the limit of the difference of the two functions as x approaches infinity is zero.

An even better approximation, and an estimate of the error term, is given by the formula

\pi(x)={\rm Li} (x) + O \left(x e^{-\frac{1}{15}\sqrt{\ln(x)}}\right)

for x → ∞ (see big O notation). Here Li(x) is the offset logarithmic integral function.

Here is a table that shows how the three functions (π(x), x/ln(x) and Li(x)) compare:

x π(x) π(x) – x/ln(x) Li(x) – π(x) x/π(x)
101 4 0  2 2.500
102 25 3  5 4.000
103 168 23  10 5.952
104 1,229 143  17 8.137
105 9,592 906  38 10.430
106 78,498 6,116  130 12.740
107 664,579 44,159  339 15.050
108 5,761,455 332,774  754 17.360
109 50,847,534 2,592,592  1,701 19.670
1010 455,052,511 20,758,029  3,104 21.980
1011 4,118,054,813 169,923,159  11,588 24.280
1012 37,607,912,018 1,416,705,193  38,263 26.590
1013 346,065,536,839 11,992,858,452  108,971 28.900
1014 3,204,941,750,802 102,838,308,636  314,890 31.200
1015 29,844,570,422,669 891,604,962,452  1,052,619 33.510
1016 279,238,341,033,925 7,804,289,844,392  3,214,632 35.810
4 ·1016 1,075,292,778,753,150 28,929,900,579,949  5,538,861 37.200

As a consequence of the prime number theorem, one get an asymptotic expression for the nth prime number p(n):

p(n)\sim n\ln(n).

One can also derive the probability that a random number n is prime: 1/ln(n).

The theorem was conjectured by Adrien-Marie Legendre in 1798 and proved independently by Hadamard and de la Vallée Poussin in 1896. The proof used methods from complex analysis, specifically the Riemann zeta function. Nowadays, so-called “elementary” proofs are available that only use number theoretic means. The first of these was provided partly independently by Paul Erdös and Atle Selberg in 1949 although it was previously believed that such proofs with only real variables could not be found.

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