Prime counting function

From Wikipedia, the free encyclopedia

In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by $π(x)$ (this does not refer to the number π).

The 60 first values of π(n)

1 History
2 Table of π(x), x / ln x, and Li(x)
3 Algorithms for evaluating π(x)
4 Other prime counting functions
5 Formulas for prime counting functions
6 Inequalities
7 The Riemann hypothesis
8 Relation to prime sums
9 References

[edit] History

Of great interest in number theory is the growth rate of the prime counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately

$x/\operatorname{ln}(x),\,$

in the sense that

$\lim_{x\rightarrow +\infty}\frac{\pi(x)}{x/\operatorname{ln}(x)}=1.\,$

This statement is the prime number theorem. An equivalent statement is

$\lim_{x\rightarrow +\infty}\pi(x) / \operatorname{li}(x)=1,\,$

where li is the logarithmic integral function. This was first proved around 1896 by Hadamard and by de la Vallée Poussin (independently), using properties of the Riemann zeta function introduced by Riemann in 1859.

More precise estimates of $π(x)$ are now known; for example

$\pi(x) = \operatorname{li}(x) + O \left( x \exp \left( -\frac{\sqrt{\ln(x)}}{15} \right) \right),$

where the O is big O notation. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).

Another conjecture about the growth rate for prime series involving the prime number theorem is

$G(n,x)= \sum_{p}^{x} p^{n} \sim \pi(x^{n+1}).$

[edit] 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.

A distributed computing project computed π(x) for x ≈ 10²³, but the project was stopped after a discrepancy of 1 was discovered in the global check. [1]

[edit] Algorithms for evaluating π(x)

A simple way to find $π(x)$ , if $x$ is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to $x$ and then to count them.

A more elaborate way of finding $π(x)$ is due to Legendre: given $x$ , if $p 1$ , $p 2$ , …, $p k$ are distinct prime numbers, then the number of integers less than or equal to $x$ which are divisible by no $p i$ is

$\lfloor x\rfloor - \sum_{i}\left\lfloor\frac{x}{p_i}\right\rfloor + \sum_{i<j}\left\lfloor\frac{x}{p_ip_j}\right\rfloor - \sum_{i<j<k}\left\lfloor\frac{x}{p_ip_jp_k}\right\rfloor + \cdots,$

(where $\lfloor\cdot\rfloor$ denotes the floor function). This number is therefore equal to

$\pi(x)-\pi\left(\sqrt{x}\right)+1\,$

when the numbers $p_1, p_2,\dots,p_k$ are the prime numbers less than or equal to the square root of $x$ .

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating $π(x)$ . Let $p 1$ , $p 2$ , …, $p n$ be the first $n$ primes and denote by $Φ(m, n)$ the number of natural numbers not greater than $m$ which are divisible by no $p i$ . Then

$\Phi(m,n)=\Phi(m,n-1)-\Phi\left(\left[\frac{m}{p_n}\right],n-1\right),\,$

Given a natural number $m$ , if $n=\pi\left(\sqrt[3]{m}\right)$ and if $\mu=\pi\left(\sqrt{m}\right)-n$ , then

$\pi(m)=\Phi(m,n)+n(\mu+1)+\frac{\mu^2-\mu}{2}-1-\sum_{k=1}^\mu\pi\left(\frac{m}{p_{n+k}}\right).\,$

Using this approach, Meissel computed $π(x)$ , for $x$ equal to 5×10⁵, 10⁶, 10⁷, and 10⁸.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real $m$ and for natural numbers $n$ , and $k$ , $P k (m, n)$ as the number of numbers not greater than m with exactly k prime factors, all greater than $p n$ . Furthermore, set $P 0 (m, n) = 1$ . Then

$\Phi(m,n)=\sum_{k=0}^{+\infty}P_k(m,n),\,$

where the sum actually has only finitely many nonzero terms. Let $y$ denote an integer such that $\sqrt[3]{m}\le y\le\sqrt{m}$ , and set $n = π(y)$ . Then $P 1 (m, n) = π(m) - n$ and $P k (m, n) = 0$ when $k$ ≥ 3. Therefore

π(m) = Φ(m, n) + n - 1 - P 2 (m, n).

The computation of $P 2 (m, n)$ can be obtained this way:

$P_2(m,n)=\sum_{y<p\le\sqrt{m}}\left(\pi\left(\frac mp\right)-\pi(p)+1\right).\,$

On the other hand, the computation of $Φ(m, n)$ can be done using the following rules:

$\Phi(m,0)=\lfloor m\rfloor;\,$
$\Phi(m,b)=\Phi(m,b-1)-\Phi\left(\frac m{p_b},b-1\right).\,$

Using his method and an IBM 701, Lehmer was able to compute $\pi\left(10^{10}\right)$ .

Chinese mathematician Hwang Cheng, in a conference about prime number functions at the University of Bordeaux used the following identities:

$e^{(a-1)\Theta}f(x)=f(ax), \,$

$J(x)=\sum_{n=1}^{\infty}\frac{\pi(x^{1/n})}{n},$

and setting $x = e t$ , Laplace-transforming both sides and applying a geometric sum on $e n Θ$ got the expression

$\frac{1}{2{\pi}i}\int_{c-i\infty}^{c+i\infty}g(s)t^{s}\,ds = \pi(t),$

$(1-e^{\Theta})\frac{\ln \zeta(s)}{s}=g(s),$

$\Theta(s)=s\frac{d}{ds}.$

[edit] Other prime counting functions

Other prime counting functions are also used because they are more convenient to work with. One is Riemann's prime counting function, denoted $Π(x)$ or $J (x)$ . This has jumps of 1/n for prime powers pⁿ, with it taking a value half-way between the two sides at discontinuities. That added detail is because then it may be defined by an inverse Mellin transform. Formally, we may define J by

$J(x) = \frac12 \bigg(\sum_{p^n < x} \frac1n\ + \sum_{p^n \le x} \frac1n\bigg)$

where p is a prime.

We may also write

$J(x) = \sum_{n=1}^\infty \frac1n \pi(x^{1/n})$ and

$\ln( \zeta (s))=s\int_{0}^{\infty} J(x) x^{-s+1}\,dx$

Also using Möbius inversion formula then:

$\pi (x)= \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} J(x^{1/n})$

and knowing the relationship between log of the Riemann function and the von Mangoldt function $Λ$ , and the Perron formula we have

$J(x)= \sum_{2}^{x} \frac{ \Lambda (n)}{\ln(n)}$

except where we have discontinuities at prime powers.

The Chebyshev function weights primes or prime powers pⁿ by ln p:

$\theta(x)=\sum_{p\le x}\ln p,$

$\psi(x) = \frac12 \bigg(\sum_{p^n < x} \ln p\ + \sum_{p^n \le x} \ln p\bigg).$

Apart from the discontinuities at prime powers, we have

$\psi(x)=\sum_{n=1}^\infty\theta(x^{1/n})=\sum_{n\le x}\Lambda(n),$

where Λ(n) is the von Mangoldt function.

[edit] Formulas for prime counting functions

These come in two kinds, arithmetic formulas and analytic formulas. The latter are what allow us to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulas.

We have the following expression for ψ:

$\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac12 \ln(1-x^{-2}).$

Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest, and the sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part.

For J we have a more complicated formula

$J(x) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^{\rho}) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t}.$

Again, the formula is valid for x > 1, and ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term li(x) is the usual logarithmic integral; however, it is not easily describable what li means in the other terms. The best way to think about it is to consider the expression $\operatorname{li}(x^\rho)$ as an abbreviation for $\operatorname{Ei}(\rho\ln x)$ , where Ei is the analytic continuation of the exponential integral function from positive reals to the complex plane with branch cut along the negative reals.

The following simple approximation can be useful: $r o u n d (e x - A ln x + B)$ where $A = 1.007$ , $B = - .05245$ .^{[citation needed]}

An explicit integral expression involving sum over non-trivial zeros and Mertens function is:

$\pi (x) = \int_{1}^{\infty} du G(u) u^{-2}( \operatorname{li}(v) - \sum_{\rho}\operatorname{li}(v^{\rho}) - \ln 2 + \int_v^\infty \frac{dt}{t(t^2-1) \ln t})$

$x 1 / u = v (x)$

$u M'(u)=G(u)= \sum_{\rho}\frac{u^{\rho}}{\zeta ' (\rho)}+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(2n-1)! \zeta (2n+1)}( \frac{2 \pi }{u})^{2n}$ Where M(u) is the Mertens function^{[citation needed]}^{[dubious — see talk page]}

[edit] Inequalities

Here are some useful inequalities for π(x).

$\pi(x) > \frac {x} {\log x}$ for x ≥ 17.

$\pi(x) < 1.25506 \frac {x} {\log x}$ for x > 1.

$\frac {x} {\log x + 2} < \pi(x) < \frac {x} {\log x - 4}$ for x ≥ 55.

Here are some inequalities for the n^th prime, p_n.

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

The left inequality holds for n ≥ 1 and the right inequality holds for n ≥ 6.

An approximation for the n^th prime number is

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

[edit] The Riemann hypothesis

The Riemann hypothesis is equivalent to a much tighter bound on the error in the estimate for $π(x)$ , and hence to a more regular distribution of prime numbers,

$\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log{x}).$

[edit] Relation to prime sums

if we had a sum of a function over all primes :

∑	f(x)
p

and we wish to accelerate its convergence we can write it as:

$\sum_{n=1}^{\infty}(-1)^{n}(\pi(n)-\pi(n-1)+1)f(n)=2f(2)-\sum_{p}f(x)+\sum_{n=1}^{\infty}(-1)^{n}f(n)$

for the series on the left we could apply Euler transform for alternating series, providing that f(n)>f(n+1) and that the 2 series converges, it also relates an alternating series to its prime sum counterpart, the main task of using this is that we can give a good approximation using only a few values of the prime number counting function.

[edit] References

Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press, volume 1 page 234 section 8.8. ISBN 0-262-02405-5.

Marc Deléglise and Jöel Rivat, Computing $π(x)$ : The Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Mathematics of Computation, vol. 65, number 33, January 1996, pages 235–245

Dickson, Leonard Eugene (2005). History of the Theory of Numbers I: Divisibility and Primality. Dover Publications. ISBN 0-486-44232-2.

Ireland, Kenneth; Rosen, Michael (1998). A Classical Introduction to Modern Number Theory, Second edition, Springer. ISBN 0-387-97329-X.

Eric W. Weisstein, Riemann Prime Counting Function at MathWorld.

Hwang H. Cheng Prime Magic conference given at the University of Bordeaux (France) at year 2001 Démarches de la Géométrie et des Nombres de l'Université du Bordeaux

Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960.

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Categories: Articles with unsourced statements since April 2007 | All articles with unsourced statements | Accuracy disputes | Analytic number theory | Prime numbers

Prime counting function

From Wikipedia, the free encyclopedia

Contents

[edit] History

[edit] Table of π(x), x / ln x, and Li(x)

[edit] Algorithms for evaluating π(x)

[edit] Other prime counting functions

[edit] Formulas for prime counting functions

[edit] Inequalities

[edit] The Riemann hypothesis

[edit] Relation to prime sums

[edit] References

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