Characterization of how many integers are prime
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the 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 (in particular, the Riemann zeta function).
The first such distribution found is "π"("N") ~, where "π"("N") is the prime-counting function (the number of primes less than or equal to "N") and log("N") is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log("N"). Consequently, a random integer with at most 2"n" digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log("N").
Statement.
Let "π"("x") be the prime-counting function defined to be the number of primes less than or equal to x, for any real number x. For example, "π"(10) 
 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that "x" / log "x" is a good approximation to "π"("x") (where log here means the natural logarithm), in the sense that the limit of the "quotient" of the two functions "π"("x") and "x" / log "x" as x increases without bound is 1:
 formula_1
known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as
 formula_2
This notation (and the theorem) does "not" say anything about the limit of the "difference" of the two functions as x increases without bound. Instead, the theorem states that "x" / log "x" approximates "π"("x") in the sense that the relative error of this approximation approaches 0 as x increases without bound.
The prime number theorem is equivalent to the statement that the nth prime number pn satisfies
 formula_3
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as n 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:
 formula_4
As outlined below, the prime number theorem is also equivalent to
 formula_5
where ϑ and ψ are the first and the second Chebyshev functions respectively, and to
 formula_692–94
where formula_7 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 "π"("a") is approximated by the function "a" / ("A" log "a" + "B"), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, with "A" 
 1 and "B" 
 −1.08366. 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 li("x") (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of "π"("x") and "x" / log("x") 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 "ζ"("s"), for real values of the argument "s", 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 x goes to infinity of "π"("x") / ("x" / log("x")) 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 x. Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for "π"("x") were strong enough for him to prove Bertrand's postulate that there exists a prime number between "n" and 2"n" for any integer "n" ≥ 2.
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 "π"("x") 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 (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function "ζ"("s") is nonzero for all complex values of the variable s that have the form "s" 
 1 + "it" with "t" ; 0.
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 (1949). Hadamard's and de la Vallée Poussin's 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 non-elementary in the sense that 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 (or a related set such as the set of prime powers) with "weights" to arrive at a function with smoother asymptotic behavior. The most common such generalized counting function is the Chebyshev function "ψ"("x"), defined by
 formula_8 
This is sometimes written as
 formula_9
where "Λ"("n") is the von Mangoldt function, namely
 formula_10
It is now relatively easy to check that the PNT is equivalent to the claim that
 formula_11
Indeed, this follows from the easy estimates
 formula_12
and (using big O notation) for any "ε" ; 0,
 formula_13
The next step is to find a useful representation for "ψ"("x"). Let "ζ"("s") be the Riemann zeta function. It can be shown that "ζ"("s") is related to the von Mangoldt function "Λ"("n"), and hence to "ψ"("x"), via the relation
 formula_14
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 x the equation
 formula_15
holds, where the sum is over all zeros (trivial and nontrivial) 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 x (claimed to be the correct asymptotic order of "ψ"("x")) appears on the right-hand side, followed by (presumably) 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:
 formula_16
which vanishes for large x. The nontrivial zeros, namely those on the critical strip 0 ≤ Re("s") ≤ 1, can potentially be of an asymptotic order comparable to the main term x if Re("ρ") 
 1, 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 "ζ"("s") is meromorphic in the half-plane Re("s") ; 0, and is analytic there except for a simple pole at "s" 
 1, and that there is a product formula
 formula_17
for Re("s") ; 1. This product formula follows from the existence of unique prime factorization of integers, and shows that "ζ"("s") is never zero in this region, so that its logarithm is defined there and
 formula_18
Write "s" 
 "x" + "iy" ; then
 formula_19
Now observe the identity
 formula_20
so that
 formula_21
for all "x" ; 1. Suppose now that "ζ"(1 + "iy") 
 0. Certainly y is not zero, since "ζ"("s") has a simple pole at "s" 
 1. Suppose that "x" ; 1 and let x tend to 1 from above. Since formula_22 has a simple pole at "s" 
 1 and "ζ"("x" + 2"iy") 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 "ψ"("x") 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.
D. J. Newman gives a quick proof of the prime number theorem (PNT). 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 formula_23, the Chebyshev functionformula_24 is used, which is obtained by dropping some of the terms from the series for formula_23. Similar to the argument in the previous proof based on Tao's lecture, we can show that "ϑ"&amp;hairsp;&amp;hairsp;("x") ≤ "π"("x")log "x", and "ϑ"&amp;hairsp;&amp;hairsp;("x") ≥ (1 - "ɛ")("π"("x") + O("x"&amp;hairsp;1-"ɛ"))log "x" for any 0 ; "ɛ" ; 1. Thus, the PNT is equivalent to formula_26.
Likewise instead of formula_27 the function formula_28 is used, which is obtained by dropping some terms in the series for formula_27. The functions formula_30 and formula_31 differ by a function holomorphic on formula_32. Since, as was shown in the previous section, formula_22 has no zeroes on the line formula_32, formula_35 has no singularities on formula_32.
One further piece of information needed in Newman's proof, and which is the key to the estimates in his simple method, is that formula_37 is bounded. This is proved using an ingenious and easy method due to Chebyshev.
Integration by parts shows how formula_38 and formula_39 are related. For formula_40, 
 formula_41
Newman's method proves the PNT by showing the integral 
 formula_42
converges, and therefore the integrand goes to zero as formula_43, 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 formula_44 is increasing, it is easy to show in this case.
To show the convergence of formula_45, for formula_46 let 
 formula_47 and formula_48 where formula_49
then
 formula_50
which is equal to a function holomorphic on the line formula_51 .
The convergence of the integral formula_45, and thus the PNT, is proved by showing that formula_53.