Prime number theorem

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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.

1 Statement of the theorem
2 The prime counting function in terms of the logarithmic integral
3 The issue of "depth"
4 Bounds on the prime counting function
5 Approximations for the nth prime number
6 Table of π(x), x/ln(x), and Li(x)
7 Poussin's result
8 See also
9 References
10 External links

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

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

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

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

This function is related to the logarithm by

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

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

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

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

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

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

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

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

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

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

Another useful bound is

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

Approximations for the nth prime number

An approximation for the n^th prime number is

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

As a consequence of the prime number theorem, one gets an asymptotic expression for the n^th number p_n

p n ˜ n ln n .

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

$n\ \ln n + n\ln\ln n - n < p_n < n \ln n + n \ln \ln n$ 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

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

holds where $π a, b (x)$ 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.

References

Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 233 in section 8.8.
Donald Knuth, The Art of Computer Programming, volume 2, section 4.5.4, exercise 36, ISBN 0-201-89684-2.