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

Algorithms for evaluating π(x)

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

A more elaborate way of finding $\backslash pi(x)$ is due to Legendre: given $x$, if $p\_1$, $p\_2$, …, $p\_k$ are distinct prime numbers, then the number of integers smaller or equal to $x$ which are divisible by no $p\_i$ is

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

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

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

$\backslash Phi(m,n)=\backslash Phi(m,n-1)-\backslash Phi\backslash left(\backslash left*,n-1\backslash right),\backslash ,$

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

$\backslash pi(m)=\backslash Phi(m,n)+n(\backslash mu+1)+\backslash frac\{\backslash mu^2-\backslash mu\}\{2\}-1-\backslash sum\_\{k=1\}^\backslash mu\backslash pi\backslash left(\backslash frac\{m\}\{p\_\{n+k\}\}\backslash right).\backslash ,$

Using this approach, Meissel computed $\backslash pi(x)$, 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

$\backslash Phi(m,n)=\backslash sum\_\{k=0\}^\{+\backslash infty\}P\_k(m,n),\backslash ,$

where the sum actually has only finitely many nonzero terms. Let $y$ denote an integer such that $\backslash sqrt*\{m\}\backslash le\; y\backslash le\backslash sqrt\{m\}$, and set $n=\backslash pi(y)$. Then $P\_1(m,n)=\backslash pi(m)-n$ and $P\_k(m,n)=0$ when $k$ ≥ 3. Therefore

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

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

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

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

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

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

$e^\{(a-1)\backslash Theta\}f(x)=f(ax)$ :$J(x)=\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{\backslash pi(x^\{1/n\})\}\{n\}$

and setting $x=e^t$, Laplace-transforming both sides and applying a geometric sum on $e^\{n\backslash Theta\}$ got the expression:

$\backslash frac\{1\}\{2\{\backslash pi\}i\}\backslash int\_\{c-i\backslash infty\}^\{c+i\backslash infty\}g(s)t^\{s\}=\backslash pi(t)$

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

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

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, denoted $\backslash Pi(x)$ 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

where p is a prime.

We may also write

$J(x)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac1n\; \backslash pi(x^\{1/n\})$ and

$ln(\; \backslash zeta\; (s))=s\backslash int\_\{0\}^\{\backslash infty\}dxJ(x)\; x^\{-s+1\}$

and knowing the relationship between log of the Riemann function and the von Mangoldt function $\backslash Lambda$, and the Perron formula we have:

$J(x)=\; \backslash sum\_\{2\}^\{x\}\; \backslash frac\{\; \backslash Lambda\; (n)\}\{ln(n)\}$

except where we have discontinuities at prime powers, and hence π can be recovered from J by Möbius inversion.

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

$\backslash theta(x)=\backslash sum\_\{p\backslash le\; x\}\backslash ln\; p,$

Apart from the discontinuities at prime powers, we have

$\backslash psi(x)=\backslash sum\_\{n=1\}^\backslash infty\backslash theta(x^\{1/n\})=\backslash sum\_\{n\backslash le\; x\}\backslash Lambda(n),$

where Λ(n) is the von Mangoldt function.

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:

$\backslash psi(x)\; =\; x\; -\; \backslash sum\_\backslash rho\; \backslash frac\{x^\backslash rho\}\{\backslash rho\}\; -\; \backslash ln\; 2\backslash pi\; -\; \backslash frac12\; \backslash ln(1-x^\{-2\}).$

For J we have a more complicated formula

$J(x)\; =\; \backslash operatorname\{li\}(x)\; -\; \backslash sum\_\{\backslash rho\}\backslash operatorname\{li\}(x^\{\backslash rho\})\; -\; \backslash ln\; 2\; +\; \backslash int\_x^\backslash infty\; \backslash frac\{dt\}\{t(t^2-1)\; \backslash ln\; t\}.$

The following simple approximation can be useful:

round^*

where x=ln(powers of 10>2) , z=ln(x) ,A=1.007 , B= -.05245

Inequalities

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

An approximation for the n^th prime number is

$p\_n\; =\; n\; \backslash ln\; n\; +\; n\; \backslash ln\; \backslash ln\; n\; -\; n\; +\; \backslash frac\; \{n\; \backslash ln\; \backslash ln\; n\; -\; 2n\}\; \{\backslash ln\; n\}\; +$

The Riemann hypothesis

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

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

Relation to prime sums

if we had a sum of a function over all primes :$\backslash sum\_\{p\}f(x)$ and we wish to accelerate its convergence we can write it as:

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

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

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

• Jose Javier Garcia Moreta, ''An Approach to the $\backslash pi(x)$ Function in Number Theory

• 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

Analytic number theory

Fonction de compte des nombres premiers | Funzione enumerativa dei primi | Funkcja π | Število praštevil