# Prime number theorem

In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. Roughly, the prime number theorem states that the density of primes numbers around n is approximately 1 / ln(n), where ln(n) denotes the natural logarithm of n.

## Statement of the theorem

Let π(x) be the prime counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1. Using Landau notation this result can be written as

$\displaystyle \pi(x)\sim\frac{x}{\ln x}$ .

This does not mean that the limit of the difference of the two functions as x approaches infinity is zero.

Based on the tables by Anton Felkel and Jurij Vega, 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.

## The prime counting function in terms of the logarithmic integral

Gauss conjectured than an even better approximation to π(x) is given by the offset logarithmic integral function Li(x), defined by

$\displaystyle \mbox{Li}(x) = \int_2^x \frac1{\ln t} \,\mbox{d}t.$

This function is related to the logarithm by

$\displaystyle \mbox{Li}(x) = \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} = \frac{x}{\ln x} + \frac{x}{(\ln x)^2} + \frac{2x}{(\ln x)^3} + \cdots$

So, the prime number theorem can also be written as π(x) ~ Li(x). The advantage of this formulation is that the error term is less. In fact, it follows from the proof of Hadamard and de la Vallée Poussin that

$\displaystyle \pi(x)={\rm Li} (x) + O \left(x \mathrm{e}^{-a\sqrt{\ln x}}\right) \quad\mbox{as } x \to \infty$

for some positive constant a, where O(…) is the big O notation. This has been improved to

$\displaystyle \pi(x)={\rm Li} (x) + O \left(x \, \exp \left( -\frac{A(\ln x)^{3/5}}{(\ln \ln x)^{1/5}} \right) \right).$

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

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

The constant involved in the remainder term is unknown.

The logarithmic integral Li(x) is larger than π(x) for "small" values of x. However, in 1914, J. E. Littlewood proved that this is not always the case. The first value of x where π(x) exceeds Li(x) is around x = 10316; see the article on Skewes' number for more details.

## The issue of "depth"

In the first half of the twentieth century, some mathematicians felt that there exists a hierarchy of techniques in mathematics, and that the prime number theorem is a "deep" theorem, whose proof requires complex analysis. Methods with only real variables were supposed to be inadequate. G. H. Hardy was one notable member of this group.

The formulation of this belief was somewhat shaken by a proof of prime number theorem based on Wiener's tauberian theorem, though this could be circumvented by awarding Wiener's theorem "depth" itself equivalent to the complex methods. However, Paul Erdős and Atle Selberg found a so-called "elementary" proof of the prime number theorem in 1949, which uses only number-theoretic means. The Selberg-Erdős work effectively laid rest to the whole concept of "depth", 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.

## Bounds on the prime counting function

The prime number theorem is an asymptotic result. Hence, it cannot be used to bound π(x).

However, some bounds on π(x) are known, for instance

$\displaystyle \frac{x}{\ln x} < \pi(x) < 1.25506 \, \frac{x}{\ln x}.$

The first inequality holds for all x ≥ 17 and the second one for x > 1.

Another useful bound is

$\displaystyle \frac {x}{\ln x + 2} < \pi(x) < \frac {x}{\ln x - 4} \quad\mbox{for } x \ge 55.$

## Approximations for the nth prime number

An approximation for the nth prime number is

$\displaystyle p_n = n \ln n + n \ln \ln n - n + \frac {n \ln \ln n - 2n} {\ln n} + O\left( \frac {n (\ln \ln n)^2} {(\ln n)^2}\right)$

As a consequence of the prime number theorem, one gets an asymptotic expression for the nth number pn

$\displaystyle p_n \sim n \ln n.$

This can be stated more precisely as a pair of bounds:

$\displaystyle n\ \ln n + n\ln\ln n - n < p_n < n \ln n + n \ln \ln n$ for n ≥ 6.

The left inequality is due to Pierre Dusart (1999) and is valid for n ≥ 2.

## Table of π(x), x/ln(x), and Li(x)

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
1017 2,623,557,157,654,233
1018 24,739,954,287,740,860
1019 234,057,667,276,344,607
1020 2,220,819,602,560,918,840
1021 21,127,269,486,018,731,928
1022 201,467,286,689,315,906,290
1023 1,925,320,391,606,818,006,727

The first column is sequence A006880 in OEIS; the second column is sequence A057835; and the third column is sequence A057752.

## Poussin's result

Poussin proved that

$\displaystyle \pi_{a,b}(x) \sim \frac{1}{\phi(a)}\mathrm{Li}(x)$

holds where $\displaystyle \pi_{a,b}(x)$ is the number of primes in the progression an+b ,and φ(n) is the Euler's totient function. This was conjectured by Dirichlet and Legendre.