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 (x) (unrelated to the number ).
thumb|right|400px|The values of (n) for the first 60 positive integers 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
\frac x {\log(x)}
in the sense that
\lim_{x\rightarrow\infty} \frac{\pi(x)}{x/\log(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.
The prime number theorem 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.
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).
In 1899, de la Vallée Poussin proved that (see also Theorem 23 of)
\pi(x) = \operatorname{li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty
for some positive constant .
Here,  is the big  notation.
More precise estimates of \pi(x)\!
are now known.
For example, in 2002, Kevin Ford proved that
\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -0.2098(\log x)^\frac35 (\log \log x)^{-\frac 1 5} \right) \right).
Mossinghoff and Trudgian proved an explicit upper bound for the difference between \pi(x) and \operatorname{li}(x):
\big| \pi(x) - \operatorname{li}(x) \big| \le 0.2593 \frac{x}{(\log x)^{3/4}} \exp \left( -\sqrt{ \frac{\log x}{6.315} } \right)
for x \ge 229.
For most values of x we are interested in (i.e., when x is not unreasonably large) \operatorname{li}(x) is greater than \pi(x).
However,  \pi(x) - \operatorname{li}(x) is known to change sign infinitely many times.
For a discussion of this, see Skewes' number.
Exact form
For x>1 let \pi_0 (x)=\pi(x)-1/2 when x is a prime number, and \pi_0 (x)=\pi(x) otherwise.
Of profound importance, Bernhard Riemann proved that \pi_0(x) is equal to
\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho)
where
\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}(x^{1/n}),
is the Möbius function,  is the logarithmic integral function, ρ indexes every zero of the Riemann zeta function, and  is not evaluated with a branch cut but instead considered as  where  is the exponential integral.
If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then \pi_0(x) may be approximated by
\pi_0(x) \approx \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho) - \frac{1}{\log{x}} + \frac{1}{\pi} \arctan{\frac{\pi}{\log{x}}} .
The Riemann hypothesis suggests that every such non-trivial zero lies along .
Table of {{pi}}(''x''), ''x'' / log ''x'', and li(''x'')
The table shows how the three functions (x), x / log x and li(x) compare at powers of 10.
See also, and
{| class="wikitable" style="text-align: right"
thumb|300px|Graph showing ratio of the prime-counting function (x) to two of its approximations, x/log x and Li(x).
As x increases (note x axis is logarithmic), both ratios tend towards 1.
The ratio for x/log x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.
In the On-Line Encyclopedia of Integer Sequences, the (x) column is sequence ,  is sequence , and  is sequence .
The value for (1024) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.
It was later verified unconditionally in a computation by D. J. Platt.
The value for (1025) is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.
The value for (1026) was computed by D. B. Staple.
All other prior entries in this table were also verified as part of that work.
The value for 1027 was announced in 2015 by David Baugh and Kim Walisch.
The value for 1028 was announced in 2020 by David Baugh and Kim Walisch.
Algorithms for evaluating {{pi}}(''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 (using the inclusion–exclusion principle): given x, if p_1,p_2,\ldots,p_n 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{x}\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,\ldots,p_n are the prime numbers less than or equal to the square root of x.
The Meissel–Lehmer algorithm
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, \ldots, 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)-\Phi\left(\frac m {p_n},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 \pi(x), for x equal to 5, 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=\pi(y).
Then P_1(m,n)=\pi(m)-n and P_k(m,n)=0 when k \geq 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:
P_2(m,n) = \sum_{y < p\le\sqrt{m}}\left(\pi\left(\frac m p \right)-\pi(p)+1\right),
where the sum is over prime numbers.
On the other hand, the computation of \Phi(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).
Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat.
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) or J_0(x).
This has jumps of 1/n for prime powers pn, with it taking a value halfway between the two sides at discontinuities.
That added detail is used because then the function may be defined by an inverse Mellin transform.
Formally, we may define \Pi_0(x) by
\Pi_0(x) = \frac 1 2 \left( \sum_{p^n < x} \frac 1 n \ + \sum_{p^n \le x} \frac 1 n \right)
where p is a prime.
We may also write
\Pi_0(x) = \sum_{n=2}^x \frac{\Lambda(n)}{\log n} - \frac 1 2 \frac{\Lambda(x)}{\log x} = \sum_{n=1}^\infty \frac 1 n \pi_0\bigl(x^{1/n}\bigr)
where \Lambda(n) is the von Mangoldt function and
\pi_0(x) = \lim_{\varepsilon \to 0} \frac{\pi(x-\varepsilon) + \pi(x+\varepsilon)} 2.
The Möbius inversion formula then gives
\pi_0(x) = \sum_{n=1}^\infty \frac{\mu(n)}n \Pi_0\bigl(x^{1/n}\bigr)
Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function \Lambda, and using the Perron formula we have
\log \zeta(s) = s \int_0^\infty \Pi_0(x) x^{-s-1} \,dx
The Chebyshev function weights primes or prime powers pn by log(p):
\theta(x) = \sum_{p\le x} \log p
\psi(x) = \sum_{p^n \le x} \log p = \sum_{n=1}^\infty \theta\bigl(x^{1/n}\bigr) = \sum_{n \le x}\Lambda(n).
Formulas for prime-counting functions
Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas.
Analytic formulas for prime-counting were the first used 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(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log 2\pi - \frac{1}{2} \log\left(1-x^{-2}\right),
where
\psi_0(x) = \lim_{\varepsilon \to 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 \Pi_0(x) we have a more complicated formula
\Pi_0(x) = \operatorname{li}(x) - \sum_{\rho} \operatorname{li}(x^\rho) - \log 2 + \int_x^\infty \frac{dt}{t \left(t^2 - 1\right) \log t}.
thumb|Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function
Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value.
The integral is equal to the series over the trivial zeros:
\int_x^\infty \frac{\mathrm dt}{t \left(t^2 - 1\right) \log t}=\int_x^\infty \frac{1}{t\log t} (\sum_{m}t^{-2m})\,\mathrm dt=\sum_{m}\int_x^\infty \frac{t^{-2m}}{t\log t} \,\mathrm dt \,\,\overset{(u=t^{-2m})}{=}-\sum_{m} \operatorname{li}(x^{-2m})
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 log x), where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals.
Thus, Möbius inversion formula gives us
\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho) - \sum_{m} \operatorname{R}(x^{-2m})
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{(\log x)^k}{k! k \zeta(k+1)}
is Riemann's R-function and  is the Möbius function.
The latter series for it is known as Gram series.
Because \log(x) < x for all x > 0, this series converges for all positive x by comparison with the series for e^x.
The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as \rho\log x   and not \log x^\rho .
thumb|right|220px|Δ-function (red line) on log scaleThe sum over non-trivial zeta zeros in the formula for \pi_0(x) describes the fluctuations of \pi_0(x), while the remaining terms give the "smooth" part of prime-counting function, so one can use
\operatorname{R}(x) - \sum_{m=1}^\infty \operatorname{R}(x^{-2m})
as a good estimator of \pi(x) for x > 1.
In fact, since the second term converges to 0, while the amplitude of the "noisy" part is heuristically about \sqrt{x}/\log x, estimating \pi(x) by
\operatorname{R}(x)
alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function
\bigl( \pi_0(x) - \operatorname{R}(x)\bigr) \frac{\log x}{\sqrt x}.
An extensive table of the values of the virtually identical function \Delta(x)=\left( \pi_0(x) - \operatorname{R}(x) + \tfrac{1}{\log x} - \tfrac{1}{\pi} \arctan\tfrac{\pi}{\log x} \right) \tfrac{\log x}{\sqrt x} is available.
Here, the extra terms \tfrac{1}{\log x} - \tfrac{1}{\pi} \arctan\tfrac{\pi}{\log x}=O\bigl((\log x)^{-3}\bigr) come from an approximation to \sum_m\operatorname R(x^{-2m}) due to Riesel and Göhl.
Inequalities
Here are some useful inequalities for (x).
\frac x {\log x} < \pi(x) < 1.25506 \frac x {\log x}
for x ≥ 17.
The left inequality holds for x ≥ 17 and the right inequality holds for x > 1.
The constant 1.25506 is \frac{30 \log 113}{113} to 5 decimal places, as \frac{\pi(x) \log x}{x} has its maximum value at x = 113.
Pierre Dusart proved in 2010:
\frac {x} {\log x - 1} < \pi(x) for x \ge 5393, and
\pi(x) <  \frac {x} {\log x - 1.1} for x \ge 60184.
Here are some inequalities for the nth  prime, pn.
The upper bound is due to Rosser (1941), the lower one to Dusart (1999):
n (\log (n \log n) - 1) < p_n < n {\log (n \log n)}  for n ≥ 6.
The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6.
An approximation for the nth prime number is
p_n = n (\log (n \log n) - 1) + \frac {n (\log \log n - 2)} {\log n} +  O\left( \frac {n (\log \log n)^2} {(\log n)^2}\right).
Ramanujan proved that the inequality
\pi(x)^2 < \frac{ex}{\log x} \pi\left( \frac{x}{e} \right)
holds for all sufficiently large values of x.
In  Dusart proved (Proposition 6.6) that, for n \ge 688383,
p_n \le n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} \right),
and (Proposition 6.7) that, for n \ge 3,
p_n \ge n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2.1}{\log n} \right) .
More recently, Dusart has proved (Theorem 5.1) that, for x > 1,
\pi(x) \le \frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} + \frac{7.59}{\log^3 x} \right) ,
and that, for x \ge 88789,
\pi(x) > \frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} \right) .
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}).
Specifically,
|\pi(x) - \operatorname{li}(x)| < \frac{\sqrt{x}}{8\pi} \, \log{x}, \qquad \text{for all } x \ge 2657.
See also
Foias constant
Bertrand's postulate
Oppermann's conjecture
Ramanujan prime
References
External links
Chris Caldwell, The Nth Prime Page at The Prime Pages.
Tomás Oliveira e Silva, Tables of prime-counting functions.
