Prime-counting function

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number "x" cite book |first=Eric |last=Bach |coauthors=Shallit, Jeffrey |year=1996 |title=Algorithmic Number Theory |publisher=MIT Press |id=ISBN 0-262-02405-5 |pages=volume 1 page 234 section 8.8] MathWorld |title=Prime Counting Function |urlname=PrimeCountingFunction] . It is denoted by $scriptstylepi(x)$ (this does "not" refer to the number π).

History

Of great interest in number theory is the growth rate of the prime-counting function cite book | first=Leonard Eugene | last=Dickson | year=2005 | title=History of the Theory of Numbers I: Divisibility and Primality | publisher=Dover Publications | id=ISBN 0-486-44232-2] . 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

:$lim\_\{x\; ightarrowinfty\}frac\{pi(x)\}\{x/operatorname\{ln\}(x)\}=1.!$

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

:$lim\_\{x\; ightarrowinfty\}pi(x)\; /\; operatorname\{li\}(x)=1!$

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

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

:$pi(x)\; =\; operatorname\{li\}(x)\; +\; O\; left(\; x\; exp\; left(\; -frac\{sqrt\{ln(x)\{15\}\; ight)\; ight)!$

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) cite book | first=Kenneth | last=Ireland | coauthors=Rosen, Michael | year=1998 | title=A Classical Introduction to Modern Number Theory | edition=Second edition | publisher=Springer | id=ISBN 0-387-97329-X ] .

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

:$sum\_\{p\; le\; x\}\; p^\{n\}\; sim\; pi(x^\{n+1\})\; sim\; Li(x^\{n+1\}).$

Table of π("x"), "x" / ln "x", and li("x")

The table shows how the three functions π("x"), "x" / ln "x" and li("x") compare at powers of 10. See also cite web |title=Tables of values of pi(x) and of pi2(x) |url=http://www.ieeta.pt/~tos/primes.html |publisher=Tomás Oliveira e Silva |accessdate=2008-09-14] , cite web |title=Values of π(x) and Δ(x) for different x's |url=http://www.primefan.ru/stuff/primes/table.html |publisher=Andrey V. Kulsha |accessdate=2008-09-14] and cite web |title=A table of values of pi(x) |url=http://numbers.computation.free.fr/Constants/Primes/pixtable.html |publisher=Xavier Gourdon, Pascal Sebah, Patrick Demichel |accessdate=2008-09-14] .

:

The π("x") column is sequence [http://www.research.att.com/projects/OEIS?Anum=A006880 A006880] in OEIS; π("x") - "x" / ln "x" is sequence [http://www.research.att.com/projects/OEIS?Anum=A057835 A057835] ; and li("x") − π("x") is sequence [http://www.research.att.com/projects/OEIS?Anum=A057752 A057752] . The value for π(10²³) is by T. O. e Silva .

Algorithms for evaluating π("x")

A simple way to find $pi(x)$, 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 $pi(x)$ 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

:

(where $lfloorcdot\; floor$ denotes the floor function). This number is therefore equal to

:$pi(x)-pileft(sqrt\{x\}\; ight)+1,$

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 $pi(x)$. Let $p\_1$, $p\_2$, …, $p\_n$ be the first $n$ primes and denote by $Phi(m,n)$ the number of natural numbers not greater than $m$ which are divisible by no $p\_i$. Then

:$Phi(m,n)=Phi(m,n-1)-Phileft(left\; [frac\{m\}\{p\_n\}\; ight]\; ,n-1\; ight).,$

Given a natural number $m$, if $n=pileft(sqrt\; [3]\; \{m\}\; ight)$ and if $mu=pileft(sqrt\{m\}\; ight)-n$, then

:$pi(m)=Phi(m,n)+n(mu+1)+frac\{mu^2-mu\}\{2\}-1-sum\_\{k=1\}^mupileft(frac\{m\}\{p\_\{n+k\; ight).,$

Using this approach, Meissel computed $pi(x)$, 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 $sqrt\; [3]\; \{m\}le\; ylesqrt\{m\}$, and set $n=pi(y)$. Then $P\_1(m,n)=pi(m)-n$ and $P\_k(m,n)=0$ when $k$ ≥ 3. Therefore

:$pi(m)=Phi(m,n)+n-1-P\_2(m,n).$

The computation of $P\_2(m,n)$ can be obtained this way:

:

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

#$Phi(m,0)=lfloor\; m\; floor;,$
#$Phi(m,b)=Phi(m,b-1)-Phileft(frac\; m\{p\_b\},b-1\; ight).,$

Using his method and an IBM 701, Lehmer was able to compute $pileft(10^\{10\}\; ight)$.

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat cite web |title=Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method |url=http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00674-6/S0025-5718-96-00674-6.pdf |publisher=Marc Deléglise and Jöel Rivat, "Mathematics of Computation", vol. 65, number 33, January 1996, pages 235–245 |accessdate=2008-09-14] .

The Chinese mathematician Hwang Cheng, in a conference about prime number functions at the University of Bordeauxcite |author=Hwang H., Cheng |title=Démarches de la Géométrie et des Nombres de l'Université du Bordeaux |publisher="Prime Magic" conference |date=2001] , 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 $e^\{nTheta\}$ got the expression

:$frac\{1\}\{2\{pi\}i\}int\_\{c-iinfty\}^\{c+iinfty\}g(s)t^\{s\},ds\; =\; pi(t),$ :$frac\{ln\; zeta(s)\}\{s\}=(1-e^\{Theta(s)\})^\{-1\}g(s)$

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

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, usually denoted as $Pi\_0(x)$. This has jumps of "1/n" for prime powers "p"^"n", 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 $Pi\_0(x)$ by

:

where "p" is a prime.

We may also write

:$Pi\_0(x)\; =\; sum\_2^x\; frac\{Lambda(n)\}\{ln\; n\}\; -\; frac12\; frac\{Lambda(x)\}\{ln\; x\}\; =\; sum\_\{n=1\}^infty\; frac1n\; pi\_0(x^\{1/n\})$

where Λ("n") is the von Mangoldt function and

:$pi\_0(x)\; =\; lim\_\{varepsilon\; ightarrow\; 0\}frac\{pi(x-varepsilon)+pi(x+varepsilon)\}2.$

Möbius inversion formula then gives

:$pi\_\{0\}(x)\; =\; sum\_\{n=1\}^infty\; frac\{mu(n)\}n\; Pi\_0(x^\{1/n\})$

Knowing the relationship between log of the Riemann zeta function and the von Mangoldt function $Lambda$, and using the Perron formula we have

:$ln\; zeta(s)\; =\; s\; int\_0^infty\; Pi\_0(x)\; x^\{-s+1\},dx$

The Chebyshev function weights primes or prime powers "p"^"n" by ln("p"):

:$heta(x)=sum\_\{ple\; x\}ln\; p$:$psi(x)\; =\; sum\_\{p^n\; le\; x\}\; ln\; p\; =\; sum\_\{n=1\}^infty\; heta(x^\{1/n\})\; =\; sum\_\{nle\; x\}Lambda(n).$

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 cite book |first=E.C. |last=Titchmarsh |year=1960 |title=The Theory of Functions, 2nd ed. |publisher=Oxford University Press] .

We have the following expression for ψ:

:$psi\_0(x)\; =\; x\; -\; sum\_\; ho\; frac\{x^\; ho\}\{\; ho\}\; -\; ln\; 2pi\; -\; frac12\; ln(1-x^\{-2\})$

where

: $psi\_0(x)\; =\; lim\_\{varepsilon\; ightarrow\; 0\}frac\{psi(x-varepsilon)+psi(x+varepsilon)\}2.$

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. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula.

For $scriptstylePi\_0(x)$ we have a more complicated formula

:$Pi\_0(x)\; =\; operatorname\{li\}(x)\; -\; sum\_\{\; ho\}operatorname\{li\}(x^\{\; ho\})\; -\; ln\; 2\; +\; int\_x^infty\; frac\{dt\}\{t(t^2-1)\; ln\; t\}.$

Again, the formula is valid for "x" > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value, and, again, the latter integral, taken with minus sign, is just the same sum, but over the trivial zeros. The first term li("x") is the usual logarithmic integral function; the expression li("x"^ρ) in the second term should be considered as Ei(ρ 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.

Thus, Möbius inversion formula gives us

:$pi\_\{0\}(x)\; =\; operatorname\{R\}(x)\; -\; sum\_\{\; ho\}operatorname\{R\}(x^\{\; ho\})\; -\; frac1\{ln\; x\}\; +\; frac1pi\; arctan\; fracpi\{ln\; x\}$

valid for "x" > 1, where

:$operatorname\{R\}(x)\; =\; sum\_\{n=1\}^\{infty\}\; frac\{\; mu\; (n)\}\{n\}\; operatorname\{li\}(x^\{1/n\})\; =\; 1\; +\; sum\_\{k=1\}^infty\; frac\{(ln\; x)^k\}\{k!\; k\; zeta(k+1)\}$

is so-called Riemann's R-function MathWorld |title=Riemann Prime Counting Function |urlname=RiemannPrimeCountingFunction] . The latter series for it is known as Gram series MathWorld|title=Gram Series |urlname=GramSeries] and converges for all positive "x".

The sum over non-trivial zeta zeros in the formula for $scriptstylepi\_0(x)$ describes the fluctuations of $scriptstylepi\_0(x)$, while the remaining terms give the "smooth" part of prime-counting function cite web |title=The encoding of the prime distribution by the zeta zeros |url=http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm |publisher=Matthew Watkins |accessdate=2008-09-14] , so one can use

:$operatorname\{R\}(x)\; -\; frac1\{ln\; x\}\; +\; frac1pi\; arctan\; fracpi\{ln\; x\}$

as the [http://primefan.ru:8014/WWW/stuff/primes/best_estimator.gifbest estimator] of $scriptstylepi(x)$ for "x" > 1.

The amplitude of the "noisy" part is heuristically about $scriptstylesqrt\; x/ln\; x$, so the fluctuations of the distribution of primes may be clearly represented with the Δ-function:

:$Delta(x)\; =\; left(\; pi\_0(x)\; -\; operatorname\{R\}(x)\; +\; frac1\{ln\; x\}\; -\; frac1\{pi\}arctanfrac\{pi\}\{ln\; x\}\; ight)\; frac\{ln\; x\}\{sqrt\; x\}.$

An extensive table of the values of Δ("x") is available .

Inequalities

Here are some useful inequalities for π("x").

: for x > 1.

: for x ≥ 55.

Here are some inequalities for the "n"^th prime, "p"_"n".Fact|date=May 2008: 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:$p\_n\; =\; n\; ln\; n\; +\; n\; ln\; ln\; n\; -\; n\; +\; frac\; \{n\; ln\; ln\; n\; -\; 2n\}\; \{ln\; n\}\; +\; Oleft(\; frac\; \{n\; (ln\; ln\; n)^2\}\; \{(ln\; n)^2\}\; ight).$

The Riemann hypothesis

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

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

Relation to prime sums

if we had a sum of a function over all primes :$sum\_\{p\}^\{\}\; f(x)\; !$ 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.

References

External links

*Chris Caldwell, [http://primes.utm.edu/nthprime/ "The Nth Prime Page"] at The Prime Pages.