Prime number theorem

In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. For any positive real number x, define

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

The prime number theorem then states that

$\pi (x)\approx {\frac {x}{\ln(x)}}$

where ln(x) is the natural logarithm of x. This notation means only 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

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

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.

Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

Helge von Koch in 1901 showed that more specifically, if the Riemann hypothesis is true, the error term in the above relation can be improved to

$\pi (x)={\rm {Li}}(x)+O\left({\sqrt {x}}\ln(x)\right)$

The constant involved in the O-notation is unknown.

The issue of 'depth'

So-called "elementary proofs" of PNT are available that only use number-theoretic means. The first of these was provided partly independently by Paul Erdös and Atle Selberg in 1949. It was previously believed by some experts in the field that such proofs could not be found. That is, it was asserted, notably by G. H. Hardy, that complex analysis was essentially involved in PNT, leading to a conception of depth of theorems. Methods with only real variables were supposed to be inadequate. This was not a rigorous, logical concept (and indeed could not be), but was rather based on the feeling that such a hierarchy of techniques should exist (for reasons of aesthetics, presumably, in Hardy's case). The formulation of this belief was somewhat shaken by a proof of PNT based on Wiener's tauberian theorem, though this could be circumvented by awarding Wiener's theorem 'depth' itself equivalent to the complex methods.

The Selberg-Erdös work effectively put paid to the whole concept, showing that technically elementary methods (in other words combinatorics) were sharper than previously expected. Subsequent development of sieve methods showed they had a definite role in prime number theory.