in the sense that
This statement is the prime number theorem. An equivalent statement is
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 are now known; for example
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
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)|
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
(where denotes the floor function). This number is therefore equal to
when the numbers are the prime numbers less than or equal to the square root of .
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:
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 .
and setting , Laplace-transforming both sides and applying a geometric sum on got the expression
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 . 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 by
where p is a prime.
We may also write
where Λ(n) is the von Mangoldt function and
Möbius inversion formula then gives
The Chebyshev function weights primes or prime powers pn by ln(p):
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:
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(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
valid for x > 1, where
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 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(x) is available .
Here are some inequalities for the nth prime, pn.
An approximation for the nth prime number is
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,
if we had a sum of a function over all primes
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.
PRIME POWERS PARKER THE RUNNING BACK ENDS MIDDLETON'S SEASON WITH 216 RUSHING YARDS AND FOUR TOUCHDOWNS.(SPORTS)
Nov 04, 2001; Byline: Rob Hernandez Wisconsin State Journal JANESVILLE -- Cas Prime met the defense's reputation head-on, turned upfield with a...