Prime number theorem

In mathematics, the prime number theorem describes the asymptotic distribution of prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
The first such distribution found is, where is the prime-counting function and is the natural logarithm of. This means that for large enough, the probability that a random integer not greater than is prime is very close to. In other words, the average gap between consecutive prime numbers among the first integers is roughly. Consequently, a random integer with at most digits is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime, whereas among positive integers of at most 2000 digits, about one in 4600 is prime.

Statement

Let be the prime-counting function defined to be the number of primes less than or equal to, for any real number . For example, because there are four prime numbers less than or equal to 10. The prime number theorem then states that is a good approximation to , in the sense that the limit of the quotient of the two functions and as increases without bound is 1:
known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as
This notation does not say anything about the limit of the difference of the two functions as increases without bound. Instead, the theorem states that approximates in the sense that the relative error of this approximation approaches 0 as increases without bound.
The prime number theorem is equivalent to the statement that the th prime number satisfies
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as increases without bound. For example, the th prime number is, and log rounds to, a relative error of about 6.4%.
On the other hand, the following asymptotic relations are logically equivalent:
As outlined [|below], the prime number theorem is also equivalent to
where and are the first and the second Chebyshev functions respectively, and to
where is the Mertens function.

History of the proof of the asymptotic law of prime numbers

Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that is approximated by the function where and are unspecified constants. In the second edition of his book on number theory he then made a more precise conjecture, with and Carl Friedrich Gauss considered the same question at age 15 or 16 "in the year 1792 or 1793", according to his own recollection in 1849. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral . Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of and stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
In two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function, for real values of the argument "", as in works of Leonhard Euler, as early as 1737. Chebyshev's papers predated Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit as goes to infinity of exists at all, then it is necessarily equal to one. He was able to prove unconditionally that this ratio is bounded above and below by 0.92129 and 1.10555, for all sufficiently large. Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for were strong enough for him to prove Bertrand's postulate that there exists a prime number between and for any integer.
An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, chiefly that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper that the idea to apply methods of complex analysis to the study of the real function originates. Extending Riemann's ideas, two proofs of the asymptotic law of the distribution of prime numbers were found independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year. Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function is nonzero for all complex values of the variable that have the form with
During the 20th century, the theorem of Hadamard and de la Vallée Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős. Hadamard's and original proofs are long and elaborate; later proofs introduced various simplifications through the use of Tauberian theorems but remained difficult to digest. A short proof was discovered in 1980 by the American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is not "elementary" since it uses Cauchy's integral theorem from complex analysis.

Proof sketch

Here is a sketch of the proof referred to in one of Terence Tao's lectures. Like most proofs of the PNT, it starts out by reformulating the problem in terms of a less intuitive, but better-behaved, prime-counting function. The idea is to count the primes with weights to arrive at a function with smoother asymptotic behavior. The most common such generalized counting function is the Chebyshev function, defined by
This is sometimes written as
where is the von Mangoldt function, namely
It is now relatively easy to check that the PNT is equivalent to the claim that
Indeed, this follows from the easy estimates
and for any,
The next step is to find a useful representation for. Let be the Riemann zeta function. It can be shown that is related to the von Mangoldt function, and hence to, via the relation
A delicate analysis of this equation and related properties of the zeta function, using the Mellin transform and Perron's formula, shows that for non-integer the equation
holds, where the sum is over all zeros of the zeta function. This striking formula is one of the so-called explicit formulas of number theory, and is already suggestive of the result we wish to prove, since the term appears on the right-hand side, followed by lower-order asymptotic terms.
The next step in the proof involves a study of the zeros of the zeta function. The trivial zeros −2, −4, −6, −8,... can be handled separately:
which vanishes for large. The nontrivial zeros, namely those on the critical strip, can potentially be of an asymptotic order comparable to the main term if, so we need to show that all zeros have real part strictly less than 1.

Non-vanishing on Re(''s'') = 1

To do this, we take for granted that is meromorphic in the half-plane, and is analytic there except for a simple pole at, and that there is a product formula
for. This product formula follows from the existence of unique prime factorization of integers, and shows that is never zero in this region, so that its logarithm is defined there and
Write ; then
Now observe the identity
so that
for all. Suppose now that. Certainly is not zero, since has a simple pole at. Suppose that and let tend to 1 from above. Since has a simple pole at and stays analytic, the left hand side in the previous inequality tends to 0, a contradiction.
Finally, we can conclude that the PNT is heuristically true. To rigorously complete the proof there are still serious technicalities to overcome, due to the fact that the summation over zeta zeros in the explicit formula for does not converge absolutely but only conditionally and in a "principal value" sense. There are several ways around this problem but many of them require rather delicate complex-analytic estimates. Edwards's book provides the details. Another method is to use Ikehara's Tauberian theorem, though this theorem is itself quite hard to prove. D.J. Newman observed that the full strength of Ikehara's theorem is not needed for the prime number theorem, and one can get away with a special case that is much easier to prove.

Newman's proof of the prime number theorem

gives a quick proof of the prime number theorem. The proof is "non-elementary" by virtue of relying on complex analysis, but uses only elementary techniques from a first course in the subject: Cauchy's integral formula, Cauchy's integral theorem and estimates of complex integrals. Here is a brief sketch of this proof. See for the complete details.
The proof uses the same preliminaries as in the previous section except instead of the function the Chebyshev function is used, which is obtained by dropping some of the terms from the series for Similar to the argument in the previous proof based on Tao's lecture, we can show that and for any Thus, the PNT is equivalent to
Likewise instead of the function is used, which is obtained by dropping some terms in the series for The functions and differ by a function holomorphic on Since, as was shown in the previous section, has no zeroes on the line and has no singularities on
One further piece of information needed in Newman's proof, and which is the key to the estimates in his simple method, is that is bounded. This is proved using an ingenious and easy method due to Chebyshev.
Integration by parts shows how and are related: For
Newman's method proves the PNT by showing the integral
converges, and therefore the integrand goes to zero as which is the PNT. In general, the convergence of the improper integral does not imply that the integrand goes to zero at infinity, since it may oscillate, but since is increasing, it is easy to show in this case.
To show the convergence of for let
then
which is equal to a function holomorphic on the line
The convergence of the integral and thus the PNT, is proved by showing that This involves change of order of limits since it can be written and therefore classified as a Tauberian theorem.
The difference is expressed using Cauchy's integral formula and then shown to be small for large by estimating the integrand: Fix and so that is holomorphic in the region where and and let be the boundary of that region. Since is in the interior of the region, Cauchy's integral formula gives
where is the factor introduced by Newman, which does not change the integral since is entire and
To estimate the integral, break the contour into two parts, where and Then
where Note that and hence are bounded; so let be some upper bound:
This bound, combined with the estimate for together give that the absolute value of the first integral must be The integrand over in the second integral is entire, so by Cauchy's integral theorem, the contour can be modified to a semicircle of radius in the left half-plane without changing the integral, and the same argument as for the first integral gives the absolute value of the second integral must be Finally, letting the third integral goes to zero since and hence goes to zero on the contour. Combining the two estimates and the limit get
This holds for any so and the PNT follows.