PRIME NUMBER THEOREM : Comprehensive Encyclopedia

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.

Statement of the theorem

Graph comparing π(x), x / ln x and Li(x)

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 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.

The prime counting function in terms of the logarithmic integral

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

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

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

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

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

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

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

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

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.

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.

The prime number theorem for arithmetic progressions

Let $π n, a (x)$ 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

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

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.

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

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

$p_n \sim n \ln n.$

A better approximation is

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

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

$n \ln n + n\ln\ln n - n < p_n < n \ln n + n \ln \ln n \quad\mbox{for } n \ge 6.$

The left inequality is due to Pierre Dusart ^[1] 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)^[2]	π(x) − x / ln x^[3]	Li(x) − π(x)^[4]	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

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 N_n 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

$N_n \sim \frac{q^n}{n}.$

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

$\frac{x}{\log_q x},$

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

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

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

$q^n = \sum_{d\mid n} d N_d,$

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

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

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.

References

G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.

^ Pierre Dusart. "The kth prime is greater than k(ln k + ln ln k-1) for k>=2". Mathematics of Computation 68 (1999), pp. 411–415.
^ Number of primes < 10^n (A006880). On-Line Encyclopedia of Integer Sequences.
^ Difference between pi(10^n) and the integer nearest to 10^n / log(10^n) (A057835). On-Line Encyclopedia of Integer Sequences.
^ Difference between Li(10^n) and Pi(10^n), where Li(x) = integral of log(x) and Pi(x) = number of primes <= x (A057752). On-Line Encyclopedia of Integer Sequences.

Jeremy Avigad, Kevin Donnelly, David Gray, and Paul Raff, A formally verified proof of the prime number theorem, e-print cs. AI/0509025 in the ArXiv, 2005.
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.
Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal, vol. 1, pages 12–28, 1995.
Donald Knuth, The Art of Computer Programming, volume 2, section 4.5.4, exercise 36, ISBN 0-201-89684-2.
Lowell Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II, Mathematics of Computation, vol. 30, no. 134, pp. 337–360, 1976.
Ivan Soprounov, A Short Proof of the Prime Number Theorem for Arithmetic Progressions, 1998.

External links

Table of Primes by Anton Felkel.
Prime formulas and Prime number theorem at MathWorld.
Prime number theorem on PlanetMath
How Many Primes Are There? and The Gaps between Primes by Chris Caldwell, University of Tennessee at Martin.
Tables of prime counting functions by Tomás Oliveira e Silva

PRIME NUMBER THEOREM

Contents