Get Prime-counting Function essential facts below. View Videos or join the Prime-counting Function discussion. Add Prime-counting Function to your PopFlock.com topic list for future reference or share this resource on social media.
Function representing the number of primes less than or equal to a given number
The table shows how the three functions ?(x), x / ln x and li(x) compare at powers of 10. See also, and
?(x) − x / ln x
li(x) − ?(x)
x / ?(x)
x / ln x % Error
Graph showing ratio of the prime-counting function ?(x) to two of its approximations, x/ln x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x/ln x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.
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 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 (using the inclusion-exclusion principle): given , if are distinct prime numbers, then the number of integers less than or equal to which are divisible by no is
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 (and used) 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 5×105, 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
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 ?:
Here ? are the zeros of the Riemann zeta function in the critical strip, where the real part of ? 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
Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function
Again, the formula is valid for x > 1, while ? 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?) in the second term should be considered as Ei(? ln 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.
is Riemann's R-function and ?(n) 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 .
?-function (red line) on log scale
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 ?-function:
An extensive table of the values of ?(x) is available.
Here are some useful inequalities for ?(x).
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.