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 speaking, the prime number theorem states that if you randomly select a number nearby some large number N, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime.

In other words, the prime numbers "thin out" as one looks at larger and larger numbers, and the prime number theorem gives a precise description of exactly how much they thin out.

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Statement of the theorem

Image:PrimeNumberTheorem.png

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

<math>\pi(x)\sim\frac{x}{\ln x}</math>.

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 1796 and proved independently by Hadamard and de la Vallée Poussin in 1896. The proof used methods from complex analysis, specifically the Riemann zeta function.

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The prime counting function in terms of the logarithmic integral

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

<math> \mbox{Li}(x) = \int_2^x \frac1{\ln t} \,\mbox{d}t = \mbox{li}(x) - \mbox{li}(2). </math>

Indeed, this integral is strongly suggestive of the notion that the 'density' of primes around t should be 1/lnt. This function is related to the logarithm by the asymptotic expansion

<math> \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 </math>

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

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

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

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

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

<math> \pi(x) = {\rm Li} (x) + O\left(\sqrt x \ln x\right). </math>

The constant involved in the big O notation was estimated in 1976 by Lowell Schoenfeld: assuming the Riemann hypothesis,

<math>|\pi(x)-{\rm Li}(x)|<\frac{\sqrt x\,\ln x}{8\pi}</math>

for all x ≥ 2657. He also derived a similar bound for the Chebyshev prime counting function ψ:

<math>|\psi(x)-x|<\frac{\sqrt x\,\ln^2 x}{8\pi}</math>

for all x ≥ 73.2.

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.

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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 the 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.

Avigad et al. (2005) contains a computer verified version of this elementary proof in the Isabelle theorem prover.

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The prime number theorem for arithmetic progressions

Let <math>\pi_{n,a}(x)</math> denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre conjectured, and Vallée Poussin proved, that, if a and n are coprime, then

<math>

\pi_{n,a}(x) \sim \frac{1}{\phi(n)}\mathrm{Li}(x), </math> where φ(·) is the Euler's totient function. In other words, the primes are distributed evenly among the residue classes [a] modulo n with gcd(a, n) = 1.

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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

<math> \frac{x}{\ln x} < \pi(x) < 1.25506 \, \frac{x}{\ln x}. </math>

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

Another useful bound is

<math>

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

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Approximations for the nth prime number

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

<math>p_n \sim n \ln n.</math>

A better approximation is

<math> p_n = n \ln n + n \ln \ln n + \frac{n}{\ln n} \big( \ln \ln n - \ln n- 2 \big)

+ O\left( \frac {n (\ln \ln n)^2} {(\ln n)^2}\right). </math>

Rosser's theorem states that pn is larger than n ln n. This can be improved by the following pair of bounds:

<math> n \ln n + n\ln\ln n - n < p_n < n \ln n + n \ln \ln n \quad\mbox{for } n \ge 6. </math>

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

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Gaps between primes

The prime number theorem says that the "average" length of the gap between a prime p and the next prime is ln p. Of course, the actual length of the gap might be much more or less than this. For instance, the gap between a pair of twin primes contains only one number. On the other hand, the error term in the prime number theorem implies an upper bound on the length of a gap: for every ε > 0, there is a number q such that the gap is smaller than εp for all primes p larger than q. This result has been improved steadily. Baker et al. (2001) proved that gap is at most p0.525 for sufficiently large p.

Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap g(p) satisfies

<math> g(p) = O(\sqrt{p} \ln p). </math>

Later, he conjectured that the gaps are even smaller, namely

<math> g(p) = O\left((\ln p)^2\right). </math>

At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

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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)
10 4 −0.3 2.2 2.500
102 25 3.3 5.1 4.000
103 168 23 10 5.952
104 1,229 143 17 8.137
105 9,592 906 38 10.425
106 78,498 6,116 130 12.740
107 664,579 44,158 339 15.047
108 5,761,455 332,774 754 17.357
109 50,847,534 2,592,592 1,701 19.667
1010 455,052,511 20,758,029 3,104 21.975
1011 4,118,054,813 169,923,159 11,588 24.283
1012 37,607,912,018 1,416,705,193 38,263 26.590
1013 346,065,536,839 11,992,858,452 108,971 28.896
1014 3,204,941,750,802 102,838,308,636 314,890 31.202
1015 29,844,570,422,669 891,604,962,452 1,052,619 33.507
1016 279,238,341,033,925 7,804,289,844,393 3,214,632 35.812
1017 2,623,557,157,654,233 68,883,734,693,281 7,956,589 38.116
1018 24,739,954,287,740,860 612,483,070,893,536 21,949,555 40.420
1019 234,057,667,276,344,607 5,481,624,169,369,960 99,877,775 42.725
1020 2,220,819,602,560,918,840 49,347,193,044,659,701 222,744,644 45.028
1021 21,127,269,486,018,731,928 446,579,871,578,168,707 597,394,254 47.332
1022 201,467,286,689,315,906,290 4,060,704,006,019,620,994 1,932,355,208 49.636
1023 1,925,320,391,606,818,006,727 37,083,513,766,592,669,113 7,236,148,412 51.939

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

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Analogue for irreducible polynomials over a finite field

There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem.

To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and let Nn be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from F, which cannot be written as products of polynomials of smaller degree. In this setting, these polynomials play the role of the prime numbers, since all other monic polynomials are built up of products of them. One can then prove that

<math>N_n \sim \frac{q^n}{n}.</math>

If we make the substitution x = qn, then the right hand side is just

<math>\frac{x}{\log_q x},</math>

which makes the analogy clearer. Since there are precisely qn monic polynomials of degree n (including the reducible ones), this can be rephrased as follows: if you select a monic polynomial of degree n randomly, then the probability of it being irreducible is about 1/n.

One can even prove an analogue of the Riemann hypothesis, namely that

<math>N_n = \frac{q^n}n + O\left(\frac{q^{n/2}}{n}\right).</math>

The proofs of these statements are far simpler than in the classical case. It involves a short combinatorial argument, summarised as follows. Every element of the degree n extension of F is a root of some irreducible polynomial whose degree d divides n; by counting these roots in two different ways one establishes that

<math>q^n = \sum_{d\mid n} d N_d,</math>

where the sum is over all divisors d of n. Möbius inversion then yields

<math>N_n = \frac1n \sum_{d\mid n} \mu(n/d) q^d,</math>

where μ(k) is the Möbius function. (This formula was known to Gauss.) The main term occurs for d = n, and it is not difficult to bound the remaining terms. The "Riemann hypothesis" statement depends on the fact that the largest proper divisor of n can be no larger than n/2.

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See also

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References

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External links

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