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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 . It is denoted by $scriptstylepi\left(x\right)$ (this does not refer to the number π).

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

$x/operatorname\left\{ln\right\}\left(x\right)!$

in the sense that

$lim_\left\{xrightarrowinfty\right\}frac\left\{pi\left(x\right)\right\}\left\{x/operatorname\left\{ln\right\}\left(x\right)\right\}=1.!$

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

$lim_\left\{xrightarrowinfty\right\}pi\left(x\right) / operatorname\left\{li\right\}\left(x\right)=1!$

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 $pi\left(x\right)!$ are now known; for example

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

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

$sum_\left\{p le x\right\} p^\left\{n\right\} sim pi\left(x^\left\{n+1\right\}\right) sim Li\left(x^\left\{n+1\right\}\right).$

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 , and .

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 1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 7,250,186,216 51.939

The π(x) column is sequence A006880 in OEIS; π(x) - x / ln x is sequence A057835; and li(x) − π(x) is sequence A057752 The value for π(1023) is by T. O. e Silva .

Algorithms for evaluating π(x)

A simple way to find $pi\left(x\right)$, 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 $pi\left(x\right)$ 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 $lfloorcdotrfloor$ denotes the floor function). This number is therefore equal to

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

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 $pi\left(x\right)$. Let $p_1$$p_2$, …, $p_n$ be the first $n$ primes and denote by $Phi\left(m,n\right)$ the number of natural numbers not greater than $m$ which are divisible by no $p_i$. Then

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

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

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

Using this approach, Meissel computed $pi\left(x\right)$, 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 $k$, $P_k\left(m,n\right)$ as the number of numbers not greater than m with exactly k prime factors, all greater than $p_n$. Furthermore, set $P_0\left(m,n\right)=1$. Then

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

where the sum actually has only finitely many nonzero terms. Let $y$ denote an integer such that $sqrt\left[3\right]\left\{m\right\}le ylesqrt\left\{m\right\}$, and set $n=pi\left(y\right)$. Then $P_1\left(m,n\right)=pi\left(m\right)-n$ and $P_k\left(m,n\right)=0$ when $k$ ≥ 3. Therefore

$pi\left(m\right)=Phi\left(m,n\right)+n-1-P_2\left(m,n\right).$

The computation of $P_2\left(m,n\right)$ can be obtained this way:

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

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

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

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat .

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

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

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

and setting $x=e^t$, Laplace-transforming both sides and applying a geometric sum on $e^\left\{nTheta\right\}$ got the expression

$frac\left\{1\right\}\left\{2\left\{pi\right\}i\right\}int_\left\{c-iinfty\right\}^\left\{c+iinfty\right\}g\left(s\right)t^\left\{s\right\},ds = pi\left(t\right),$
$frac\left\{ln zeta\left(s\right)\right\}\left\{s\right\}=\left(1-e^\left\{Theta\left(s\right)\right\}\right)^\left\{-1\right\}g\left(s\right)$

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

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 $Pi_0\left(x\right)$. 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 $Pi_0\left(x\right)$ by

$Pi_0\left(x\right) = frac12 bigg\left(sum_\left\{p^n < x\right\} frac1n + sum_\left\{p^n le x\right\} frac1nbigg\right)$

where p is a prime.

We may also write

$Pi_0\left(x\right) = sum_2^x frac\left\{Lambda\left(n\right)\right\}\left\{ln n\right\} - frac12 frac\left\{Lambda\left(x\right)\right\}\left\{ln x\right\} = sum_\left\{n=1\right\}^infty frac1n pi_0\left(x^\left\{1/n\right\}\right)$

where Λ(n) is the von Mangoldt function and

$pi_0\left(x\right) = lim_\left\{varepsilon rightarrow 0\right\}frac\left\{pi\left(x-varepsilon\right)+pi\left(x+varepsilon\right)\right\}2.$

Möbius inversion formula then gives

$pi_\left\{0\right\}\left(x\right) = sum_\left\{n=1\right\}^infty frac\left\{mu\left(n\right)\right\}n Pi_0\left(x^\left\{1/n\right\}\right)$

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

$ln zeta\left(s\right) = s int_0^infty Pi_0\left(x\right) x^\left\{-s+1\right\},dx$

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

$theta\left(x\right)=sum_\left\{ple x\right\}ln p$
$psi\left(x\right) = sum_\left\{p^n le x\right\} ln p = sum_\left\{n=1\right\}^infty theta\left(x^\left\{1/n\right\}\right) = sum_\left\{nle x\right\}Lambda\left(n\right).$

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:

$psi_0\left(x\right) = x - sum_rho frac\left\{x^rho\right\}\left\{rho\right\} - ln 2pi - frac12 ln\left(1-x^\left\{-2\right\}\right)$

where

$psi_0\left(x\right) = lim_\left\{varepsilon rightarrow 0\right\}frac\left\{psi\left(x-varepsilon\right)+psi\left(x+varepsilon\right)\right\}2.$

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 $scriptstylePi_0\left(x\right)$ we have a more complicated formula

$Pi_0\left(x\right) = operatorname\left\{li\right\}\left(x\right) - sum_\left\{rho\right\}operatorname\left\{li\right\}\left(x^\left\{rho\right\}\right) - ln 2 + int_x^infty frac\left\{dt\right\}\left\{t\left(t^2-1\right) ln t\right\}.$

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(xρ) 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

$pi_\left\{0\right\}\left(x\right) = operatorname\left\{R\right\}\left(x\right) - sum_\left\{rho\right\}operatorname\left\{R\right\}\left(x^\left\{rho\right\}\right) - frac1\left\{ln x\right\} + frac1pi arctan fracpi\left\{ln x\right\}$

valid for x > 1, where

$operatorname\left\{R\right\}\left(x\right) = sum_\left\{n=1\right\}^\left\{infty\right\} frac\left\{ mu \left(n\right)\right\}\left\{n\right\} operatorname\left\{li\right\}\left(x^\left\{1/n\right\}\right) = 1 + sum_\left\{k=1\right\}^infty frac\left\{\left(ln x\right)^k\right\}\left\{k! k zeta\left(k+1\right)\right\}$

is so-called Riemann's R-function . The latter series for it is known as Gram series and converges for all positive x.

The sum over non-trivial zeta zeros in the formula for $scriptstylepi_0\left(x\right)$ describes the fluctuations of $scriptstylepi_0\left(x\right)$, while the remaining terms give the "smooth" part of prime-counting function , so one can use

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

as the best estimator of $scriptstylepi\left(x\right)$ 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:

$Delta\left(x\right) = left\left(pi_0\left(x\right) - operatorname\left\{R\right\}\left(x\right) + frac1\left\{ln x\right\} - frac1\left\{pi\right\}arctanfrac\left\{pi\right\}\left\{ln x\right\} right\right) frac\left\{ln x\right\}\left\{sqrt x\right\}.$

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

Inequalities

Here are some useful inequalities for π(x).


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 nth prime, pn.


n ln n + nlnln 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 nth prime number is

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

The Riemann hypothesis

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

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

Relation to prime sums

if we had a sum of a function over all primes

$sum_\left\{p\right\}^\left\{\right\} f\left(x\right) !$
and we wish to accelerate its convergence we can write it as:

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

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.