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 .

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
in the sense that
This statement is the prime number theorem. An equivalent statement is
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.
In 1899, de la Vallée Poussin proved that
for some positive constant. Here, is the big notation.
More precise estimates of are now known. For example, in 2002, Kevin Ford proved that
In 2016, Tim Trudgian proved an explicit upper bound for the difference between and :
for.
For most values of we are interested in is greater than. However, is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

Exact form

Of profound importance, Bernhard Riemann proved that the prime-counting function is exactly
where
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. Equivalently, if the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then may be written
The Riemann hypothesis suggests that every such non-trivial zero lies along.

Table of (''x''), ''x'' / ln ''x'', and li(''x'')

The table shows how the three functions, x / ln x and li compare at powers of 10. See also, and
In the On-Line Encyclopedia of Integer Sequences, the column is sequence, is sequence, and is sequence.
The value for 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 is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.
The value for 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 published in 2015 by David Baugh and Kim Walisch.

Algorithms for evaluating (''x'')

A simple way to find, if is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to and then to count them.
A more elaborate way of finding is due to Legendre : given, if are distinct prime numbers, then the number of integers less than or equal to which are divisible by no is
. This number is therefore equal to
when the numbers are the prime numbers less than or equal to the square root of.

The Meissel–Lehmer algorithm

In a series of articles published between 1870 and 1885, Ernst Meissel described a practical combinatorial way of evaluating. Let, be the first primes and denote by the number of natural numbers not greater than which are divisible by no. Then
Given a natural number, if and if, then
Using this approach, Meissel computed, for equal to 5, 106, 107, and 108.
In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real and for natural numbers and, as the number of numbers not greater than m with exactly k prime factors, all greater than. Furthermore, set. Then
where the sum actually has only finitely many nonzero terms. Let denote an integer such that, and set. Then and when ≥ 3. Therefore,
The computation of can be obtained this way:
where the sum is over prime numbers.
On the other hand, the computation of can be done using the following rules:
Using his method and an IBM 701, Lehmer was able to compute.
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 or. 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 by
where p is a prime.
We may also write
where Λ is the von Mangoldt function and
The Möbius inversion formula then gives
Knowing the relationship between log of the Riemann zeta function and the von Mangoldt function, and using the Perron formula we have
The Chebyshev function weights primes or prime powers pn by ln:

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:
where
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 we have a more complicated formula
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 is the usual logarithmic integral function; the expression li in the second term should be considered as Ei, 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
valid for x > 1, where
is Riemann's R-function and is the Möbius function. The latter series for it is known as Gram series. Because for all, this series converges for all positive x by comparison with the series for.
The sum over non-trivial zeta zeros in the formula for describes the fluctuations of while the remaining terms give the "smooth" part of prime-counting function, so one can use
as the best estimator of for x > 1.
The amplitude of the "noisy" part is heuristically about so the fluctuations of the distribution of primes may be clearly represented with the Δ-function:
An extensive table of the values of Δ is available.

Inequalities

Here are some useful inequalities for.
for x ≥ 17.
The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is to 5 decimal places, as has its maximum value at x = 113.
Pierre Dusart proved in 2010:
Here are some inequalities for the nth prime, pn. The upper bound is due to Rosser, the lower one to Dusart :
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
Ramanujan proved that the inequality
holds for all sufficiently large values of.
In, Dusart proved that, for,
and that, for,
More recently, Dusart
has proved that, for,
and that, for,

The Riemann hypothesis

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