View source for Prime number theorem

{{short description|Theorem in number theory}}
In [[number theory]], the '''prime number theorem''' ('''PNT''') describes the [[asymptotic analysis|asymptotic]] distribution of the [[prime number]]s 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 {{math|''π''(''N'') ~ {{sfrac|''N''|log(''N'')}}}}, where {{math|''π''(''N'')}} is the [[prime-counting function]] and {{math|log(''N'')}} is the [[natural logarithm]] of {{mvar|N}}. This means that for large enough {{mvar|N}}, the [[probability]] that a random integer not greater than {{mvar|N}} is prime is very close to {{math|1 / log(''N'')}}. Consequently, a random integer with at most {{math|2''n''}} digits (for large enough {{mvar|n}}) is about half as likely to be prime as a random integer with at most {{mvar|n}} digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime ({{math|log(10<sup>1000</sup>) ≈ 2302.6}}), whereas among positive integers of at most 2000 digits, about one in 4600 is prime ({{math|log(10<sup>2000</sup>) ≈ 4605.2}}). In other words, the average gap between consecutive prime numbers among the first {{mvar|N}} integers is roughly {{math|log(''N'')}}.<ref>{{cite book|last = Hoffman|first = Paul|title = The Man Who Loved Only Numbers|url = https://archive.org/details/manwholovedonlyn00hoff|url-access = registration|publisher = Hyperion Books|year = 1998|page = 227|isbn = 978-0-7868-8406-3 | mr=1666054 | location=New York}}</ref>

==Statement==
[[File:Prime number theorem ratio convergence.svg|thumb|300px|Graph showing ratio of the prime-counting function {{math|''π''(''x'')}} to two of its approximations, {{math|''x'' / log ''x''}} and {{math|Li(''x'')}}. As {{mvar|x}} increases (note {{mvar|x}} axis is logarithmic), both ratios tend towards 1. The ratio for {{math|''x'' / log ''x''}} converges from above very slowly, while the ratio for {{math|Li(''x'')}} converges more quickly from below.]]
[[File:Prime number theorem absolute error.svg|thumb|300px|Log-log plot showing absolute error of {{math|''x'' / log ''x''}} and {{math|Li(''x'')}}, two approximations to the prime-counting function {{math|''π''(''x'')}}. Unlike the ratio, the difference between {{math|''π''(''x'')}} and {{math|''x'' / log ''x''}} increases without bound as {{mvar|x}} increases. On the other hand, {{math|Li(''x'') − ''π''(''x'')}} switches sign infinitely many times.]]
<!--[[Image:PrimeNumberTheorem.svg|thumb|right|250px|Graph comparing {{math|''π''(''x'')}} (red), {{math|''x'' / log ''x''}} (green) and {{math|Li(''x'')}} (blue)]] -->

Let {{math|''π''(''x'')}} be the [[prime-counting function]] that gives the number of primes less than or equal to {{mvar|x}}, for any real number&nbsp;{{mvar|x}}. For example, {{math|''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 {{math|''x'' / log ''x''}} is a good approximation to {{math|''π''(''x'')}}, in the sense that the [[limit of a function|limit]] of the ''quotient'' of the two functions {{math|''π''(''x'')}} and {{math|''x'' / log ''x''}} as {{mvar|x}} increases without bound is 1:

: <math>\lim_{x\to\infty}\frac{\;\pi(x)\;}{\;\left[ \frac{x}{\log(x)}\right]\;} = 1,</math>

known as '''the asymptotic law of distribution of prime numbers'''. Using [[asymptotic notation]] this result can be restated as

:<math>\pi(x)\sim \frac{x}{\log x}.</math>

This notation (and the [[theorem]]) does ''not'' say anything about the limit of the ''difference'' of the two functions as {{mvar|x}} increases without bound. Instead, the theorem states that {{math|''x'' / log ''x''}} approximates {{math|''π''(''x'')}} in the sense that the [[relative error]] of this approximation approaches 0 as {{mvar|x}} increases without bound.

The prime number theorem is equivalent to the statement that the {{mvar|n}}th prime number {{mvar|p<sub>n</sub>}} satisfies

:<math>p_n \sim n\log(n),</math>

the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as {{mvar|n}} increases without bound. For example, the {{val|2|e=17}}th prime number is {{val|8512677386048191063}},<ref>{{cite web|title=Prime Curios!: 8512677386048191063|url=http://primes.utm.edu/curios/cpage/24149.html|work=Prime Curios!|publisher=University of Tennessee at Martin|date=2011-10-09}}</ref> and ({{val|2|e=17}})log({{val|2|e=17}}) rounds to {{val|7967418752291744388}}, a relative error of about 6.4%.

The prime number theorem is also equivalent to
:<math>\lim_{x\to\infty} \frac{\vartheta (x)}x = \lim_{x\to\infty} \frac{\psi(x)}x=1,</math>
where {{mvar|ϑ}} and {{mvar|ψ}} are [[Chebyshev function|the first and the second Chebyshev functions]] respectively.

==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 {{math|''π''(''a'')}} is approximated by the function {{math|''a'' / (''A'' log ''a'' + ''B'')}}, where {{mvar|A}} and {{mvar|B}} are unspecified constants. In the second edition of his book on number theory (1808) he then made a [[Legendre's constant|more precise conjecture]], with {{math|''A'' {{=}} 1}} and {{math|''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.<ref>C. F. Gauss. ''Werke'', Bd 2, 1st ed, 444–447. Göttingen 1863.</ref> In 1838 [[Peter Gustav Lejeune Dirichlet]] came up with his own approximating function, the [[logarithmic integral]] {{math|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 {{math|''π''(''x'')}} and {{math|''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 {{math|''ζ''(''s'')}}, for real values of the argument "{{mvar|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 of {{math|''π''(''x'') / (''x'' / log(''x''))}} as {{mvar|x}} goes to infinity exists at all, then it is necessarily equal to one.<ref>{{cite journal |first=N. |last=Costa Pereira |jstor=2322510 |title=A Short Proof of Chebyshev's Theorem |journal=American Mathematical Monthly|date=August–September 1985|pages=494–495|volume=92|doi=10.2307/2322510|issue=7}}</ref> He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near 1, for all sufficiently large {{mvar|x}}.<ref>{{cite journal |first=M. |last=Nair |jstor=2320934 |title=On Chebyshev-Type Inequalities for Primes |journal=American Mathematical Monthly |date=February 1982 |pages=126–129 |volume=89 |doi=10.2307/2320934 |issue=2}}</ref> Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for {{math|''π''(''x'')}} were strong enough for him to prove [[Bertrand's postulate]] that there exists a prime number between {{math|''n''}} and {{math|2''n''}} for any integer {{math|''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, the chief of them being 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 of Riemann that the idea to apply methods of [[complex analysis]] to the study of the real function {{math|''π''(''x'')}} originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained 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 {{math|''ζ''(''s'')}} is non-zero for all complex values of the variable {{mvar|s}} that have the form {{math|''s'' {{=}} 1 + ''it''}} with {{math|''t'' > 0}}.<ref>{{cite book |last = Ingham |first = A. E. |title = The Distribution of Prime Numbers |publisher = Cambridge University Press| year = 1990 |pages = 2–5 |isbn = 978-0-521-39789-6}}</ref>

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). While the original proofs of Hadamard and de la Vallée Poussin 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 American mathematician [[Donald J. Newman]].<ref>{{cite journal|title=Simple analytic proof of the prime number theorem|journal=[[American Mathematical Monthly]] |volume=87 |year=1980 |pages=693–696 |first=Donald J. | last=Newman |doi=10.2307/2321853 |jstor=2321853 |issue=9 | mr=0602825}}</ref><ref>{{cite journal |title=Newman's short proof of the prime number theorem |journal=American Mathematical Monthly |volume=104 |year=1997 |pages=705–708 |first=Don |last=Zagier |url=http://www.maa.org/programs/maa-awards/writing-awards/newmans-short-proof-of-the-prime-number-theorem |doi=10.2307/2975232 |jstor=2975232 |issue=8 | mr=1476753}}</ref> 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<ref>{{cite web |last1=Tao |first1=Terence |authorlink = Terence Tao|title=254A, Notes 2: Complex-analytic multiplicative number theory |url=https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/ |website=Terence Tao's blog}}</ref>. 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]] {{math|''ψ''(''x'')}}, defined by

:<math>\psi(x) =\!\!\!\! \sum_\stackrel{p^k \le x,}{p \text{ is prime}} \!\!\!\!\log p.</math>

This is sometimes written as

:<math>\psi(x)=\sum_{n\le x} \Lambda(n),</math>

where {{math|''Λ''(''n'')}} is the [[von Mangoldt function]], namely

:<math>\Lambda(n) = \begin{cases} \log p & \text{if }n=p^k \text{ for some prime } p \text{ and integer } k \ge 1, \\ 0 & \text{otherwise.} \end{cases}</math>

It is now relatively easy to check that the PNT is equivalent to the claim that
:<math>\lim_{x\to\infty} \frac{\psi(x)}{x}=1.</math>
Indeed, this follows from the easy estimates
:<math>\psi(x) = \sum_{p\le x} \log p \left\lfloor \frac{\log x}{\log p} \right\rfloor \le \sum_{p\le x} \log x = \pi(x)\log x</math>
and (using [[big O notation|big {{mvar|O}} notation]]) for any {{math|''ε'' > 0}},
:<math>\psi(x) \ge \!\!\!\!\sum_{x^{1-\varepsilon}\le p\le x}\!\!\!\! \log p\ge\!\!\!\!\sum_{x^{1-\varepsilon}\le p\le x}\!\!\!\!(1-\varepsilon)\log x=(1-\varepsilon)\left(\pi(x)+O\left(x^{1-\varepsilon}\right)\right)\log x.</math>

The next step is to find a useful representation for {{math|''ψ''(''x'')}}. Let {{math|''ζ''(''s'')}} be the Riemann zeta function. It can be shown that {{math|''ζ''(''s'')}} is related to the von Mangoldt function {{math|''Λ''(''n'')}}, and hence to {{math|''ψ''(''x'')}}, via the relation

:<math>-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n = 1}^\infty \Lambda(n) n^{-s}. </math>

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 {{mvar|x}} the equation

:<math>\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log(2\pi)</math>

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 formulae (L-function)|explicit formulas of number theory]], and is already suggestive of the result we wish to prove, since the term {{mvar|x}} (claimed to be the correct asymptotic order of {{math|''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:
:<math>\sum_{n=1}^\infty \frac{1}{2n\,x^{2n}} = -\frac{1}{2}\log\left(1-\frac{1}{x^2}\right),</math>
which vanishes for a large {{mvar|x}}. The nontrivial zeros, namely those on the critical strip {{math|0 ≤ Re(''s'') ≤ 1}}, can potentially be of an asymptotic order comparable to the main term {{mvar|x}} if {{math|Re(''ρ'') {{=}} 1}}, so we need to show that all zeros have real part strictly less than 1.

To do this, we take for granted that {{math|''ζ''(''s'')}} is meromorphic in the half-plane {{math|Re(''s'') > 0}}, and is analytic there except for a simple pole at {{math|''s'' {{=}} 1}}, and that there is a product formula
:<math>\zeta(s)=\prod_p\frac{1}{1-p^{-s}} </math>
for {{math|Re(''s'') > 1}}. This product formula follows from the existence of unique prime factorization of integers, and shows that {{math|''ζ''(''s'')}} is never zero in this region, so that its logarithm is defined there and
:<math>\log\zeta(s)=-\sum_p\log \left(1-p^{-s} \right)=\sum_{p,n}\frac{p^{-ns}}{n}.</math>

Write {{math|''s'' {{=}} ''x'' + ''iy''}}; then

:<math>\big|\zeta(x+iy)\big|=\exp\left(\sum_{n,p}\frac{\cos ny\log p}{np^{nx}}\right).</math>

Now observe the identity
:<math>3+4\cos\phi+\cos2\phi=2(1+\cos\phi)^2\ge 0,</math>
so that

:<math>\left|\zeta(x)^3\zeta(x+iy)^4\zeta(x+2iy)\right|=\exp\left(\sum_{n,p}\frac{3+4\cos(ny\log p) +\cos (2ny\log p)}{np^{nx}}\right)\ge 1</math>

for all {{math|''x'' > 1}}. Suppose now that {{math|''ζ''(1 + ''iy'') {{=}} 0}}. Certainly {{mvar|y}} is not zero, since {{math|''ζ''(''s'')}} has a simple pole at {{math|''s'' {{=}} 1}}. Suppose that {{math|''x'' > 1}} and let {{mvar|x}} tend to 1 from above. Since <math>\zeta(s)</math> has a simple pole at {{math|''s'' {{=}} 1}} and {{math|''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 {{math|''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 that are beyond the scope of this paper. Edwards's book<ref>{{cite book|last = Edwards|first = Harold M.|authorlink = Harold Edwards (mathematician)|title = Riemann's zeta function |publisher = Courier Dover Publications|year = 2001|isbn = 978-0-486-41740-0}}</ref> 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.

==Prime-counting function in terms of the logarithmic integral==
In a handwritten note on a reprint of his 1838 paper "{{lang|fr|Sur l'usage des séries infinies dans la théorie des nombres}}", which he mailed to Gauss, Dirichlet conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to {{math|''π''(''x'')}} is given by the [[logarithmic integral function|offset logarithmic integral]] function {{math|Li(''x'')}}, defined by

:<math> \operatorname{Li}(x) = \int_2^x \frac{dt}{\log t} = \operatorname{li}(x) - \operatorname{li}(2). </math>

Indeed, this integral is strongly suggestive of the notion that the "density" of primes around {{mvar|t}} should be {{math|1 / log ''t''}}. This function is related to the logarithm by the [[asymptotic expansion]]

:<math> \operatorname{Li}(x) \sim \frac{x}{\log x} \sum_{k=0}^\infty \frac{k!}{(\log x)^k} = \frac{x}{\log x} + \frac{x}{(\log x)^2} + \frac{2x}{(\log x)^3} + \cdots </math>

So, the prime number theorem can also be written as {{math|''π''(''x'') ~ Li(''x'')}}. In fact, in another paper in 1899 de la Vallée Poussin proved that

:<math> \pi(x) = \operatorname{Li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math>

for some positive constant {{mvar|a}}, where {{math|''O''(...)}} is the [[big O notation|big {{mvar|O}} notation]]. This has been improved to

:<math>\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -\frac{A(\log x)^\frac35}{(\log \log x)^\frac15} \right) \right)</math> where <math>A = 0.2098</math>.<ref name="Ford">{{cite journal |author = Kevin Ford |author-link = Kevin Ford (mathematician) |title=Vinogradov's Integral and Bounds for the Riemann Zeta Function |journal=Proc. London Math. Soc. |date=2002 |volume=85 |issue = 3 |pages=565–633 |url=https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf |doi=10.1112/S0024611502013655 }}</ref>

In 2016, Trudgian proved an explicit upper bound for the difference between <math>\pi(x)</math> and <math>\operatorname{li}(x)</math>:
:<math>\big| \pi(x) - \operatorname{li}(x) \big| \le 0.2795 \frac{x}{(\log x)^{3/4}}
\exp \left( -\sqrt{ \frac{\log x}{6.455} } \right)</math>
for <math>x \ge 229</math>.<ref>{{cite journal |author = Tim Trudgian |title = Updating the error term in the prime number theorem |journal = Ramanujan Journal |volume = 39 |issue = 2  |pages=225–234  |date = February 2016 |doi = 10.1007/s11139-014-9656-6  |arxiv = 1401.2689 }}</ref>

Because of the connection between the Riemann zeta function and {{math|''π''(''x'')}}, the [[Riemann hypothesis]] has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today. More specifically, [[Helge von Koch]] showed in 1901<ref>{{cite journal|title=Sur la distribution des nombres premiers|journal=[[Acta Mathematica]]|volume=24|issue=1|year=1901|pages=159–182|first=Helge |last=Von Koch|doi=10.1007/BF02403071|language=French|trans-title=On the distribution of prime numbers|mr=1554926}}</ref> that, [[if and only if]] the Riemann hypothesis is true, the error term in the above relation can be improved to

:<math> \pi(x) = \operatorname{Li} (x) + O\left(\sqrt x \log x\right). </math>

The constant involved in the big {{mvar|O}} notation was estimated in 1976 by [[Lowell Schoenfeld]]:<ref>{{Cite journal |last=Schoenfeld |first=Lowell |title=Sharper Bounds for the Chebyshev Functions {{math|''θ''(''x'')}} and {{math|''ψ''(''x'')}}. II |journal=Mathematics of Computation |volume=30 |issue=134 |year=1976 |pages=337–360 |doi=10.2307/2005976 |jstor=2005976 | mr=0457374 }}.</ref> assuming the Riemann hypothesis,

:<math>\big|\pi(x) - \operatorname{li}(x)\big| < \frac{\sqrt x \log x}{8\pi}</math>

for all {{math|''x'' ≥ 2657}}. He also derived a similar bound for the [[Chebyshev function|Chebyshev prime-counting function]] {{mvar|ψ}}:

:<math>\big|\psi(x) - x\big| < \frac{\sqrt x (\log x)^2 }{8\pi}</math>

for all {{math|''x'' ≥ 73.2}}. This latter bound has been shown to express a variance to mean [[power law]] (when regarded as a random function over the integers), [[pink noise|{{math|{{sfrac|''f''}}}} noise]] and to also correspond to the [[Tweedie distribution|Tweedie compound Poisson distribution]]. Parenthetically, the Tweedie distributions represent a family of [[scale invariant]] distributions that serve as foci of convergence for a generalization of the [[central limit theorem]].<ref>{{cite journal|last=Jørgensen |first=Bent |last2=Martínez |first2=José Raúl |last3=Tsao |first3=Min |title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics|year=1994|volume=21|issue=3 |pages=223–243|mr=1292637 | jstor=4616314 }}</ref>

The logarithmic integral {{math|li(''x'')}} is larger than {{math|''π''(''x'')}} for "small" values of {{mvar|x}}. This is because it is (in some sense) counting not primes, but prime powers, where a power {{math|''p''<sup>''n''</sup>}} of a prime {{mvar|p}} is counted as {{math|{{sfrac|1|''n''}}}} of a prime. This suggests that {{math|li(''x'')}} should usually be larger than {{math|''π''(''x'')}} by roughly {{math|li({{sqrt|''x''}}) / 2}}, and in particular should always be larger than {{math|''π''(''x'')}}. However, in 1914, [[John Edensor Littlewood|J. E. Littlewood]] proved that <math>\pi(x) - \operatorname{li}(x)</math> changes sign infinitely often.
<ref>{{cite journal |first=J. E.|last= Littlewood | authorlink=J. E. Littlewood | title=Sur la distribution des nombres premiers|journal=[[Comptes Rendus]]|volume= 158 |year=1914|pages= 1869–1872 | jfm=45.0305.01}}</ref> The first value of {{mvar|x}} where {{math|''π''(''x'')}} exceeds {{math|li(''x'')}} is probably around {{math|''x'' {{=}} 10<sup>316</sup>}}; see the article on [[Skewes' number]] for more details. (On the other hand, the [[offset logarithmic integral]] {{math|Li(''x'')}} is smaller than {{math|''π''(''x'')}} already for {{math|''x'' {{=}} 2}}; indeed, {{math|Li(2) {{=}} 0}}, while {{math|''π''(2) {{=}} 1}}.)

==Elementary proofs==
In the first half of the twentieth century, some mathematicians (notably [[G. H. Hardy]]) believed that there exists a hierarchy of proof methods in mathematics depending on what sorts of numbers ([[integer]]s, [[real number|reals]], [[complex number|complex]]) a proof requires, and that the prime number theorem (PNT) is a "deep" theorem by virtue of requiring [[complex analysis]].<ref name="Goldfeld Historical Perspective">{{cite book | first=Dorian | last=Goldfeld | chapter-url=http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf | chapter=The elementary proof of the prime number theorem: an historical perspective | year=2004 | title=Number theory (New York, 2003) | pages=179–192 | mr=2044518 | editor1-last=Chudnovsky | editor1-first=David | editor2-last=Chudnovsky | editor2-first=Gregory | editor3-last=Nathanson | editor3-first=Melvyn | location=New York | publisher=Springer-Verlag | isbn=978-0-387-40655-8  | doi=10.1007/978-1-4419-9060-0_10}}</ref> This belief was somewhat shaken by a proof of the PNT based on [[Wiener's tauberian theorem]], though this could be set aside if Wiener's theorem were deemed to have a "depth" equivalent to that of complex variable methods.

In March 1948, [[Atle Selberg]] established, by "elementary" means, the asymptotic formula
:<math>\vartheta ( x )\log ( x ) + \sum\limits_{p \le x} {\log ( p )}\ \vartheta \left( {\frac{x}{p}} \right) = 2x\log ( x ) + O( x )</math>

where
:<math>\vartheta ( x ) = \sum\limits_{p \le x} {\log ( p )}</math>

for primes {{mvar|p}}.<ref>{{cite journal|last=Selberg|first=Atle|title=An Elementary Proof of the Prime-Number Theorem|journal=[[Annals of Mathematics]]|year=1949|volume=50|issue=2|pages=305–313|doi=10.2307/1969455|mr=0029410|jstor=1969455}}</ref> By July of that year, Selberg and [[Paul Erdős]] had each obtained elementary proofs of the PNT, both using Selberg's asymptotic formula as a starting point.<ref name="Goldfeld Historical Perspective"/><ref name=interview>{{Cite journal|url=http://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/S0273-0979-08-01223-8.pdf |first=Nils A.|last= Baas|first2= Christian F.|last2= Skau |journal= Bull. Amer. Math. Soc. |volume=45 |year=2008|pages= 617–649 |title=The lord of the numbers, Atle Selberg. On his life and mathematics|doi=10.1090/S0273-0979-08-01223-8|issue=4|mr=2434348}}</ref> These proofs effectively laid to rest the notion that the PNT was "deep", and showed that technically "elementary" methods were more powerful than had been believed to be the case. On the history of the elementary proofs of the PNT, including the Erdős–Selberg [[priority dispute]], see an article by [[Dorian Goldfeld]].<ref name="Goldfeld Historical Perspective" />

There is some debate about the significance of Erdős and Selberg's result. There is no rigorous and widely accepted definition of the notion of [[elementary proof]] in number theory, so it is not clear exactly in what sense their proof is "elementary". Although it does not use complex analysis, it is in fact much more technical than the standard proof of PNT. One possible definition of an "elementary" proof is "one that can be carried out in first order [[Peano arithmetic]]." There are number-theoretic statements (for example, the [[Paris–Harrington theorem]]) provable using [[second order arithmetic|second order]] but not [[first order arithmetic|first order]] methods, but such theorems are rare to date. Erdős and Selberg's proof can certainly be formalized in Peano arithmetic, and in 1994, Charalambos Cornaros and Costas Dimitracopoulos proved that their proof can be formalized in a very weak fragment of PA, namely {{math|''I''Δ<sub>0</sub> + exp}},<ref>.{{cite journal|last1=Cornaros|first1=Charalambos|last2=Dimitracopoulos|first2=Costas|title=The prime number theorem and fragments of ''PA''|year=1994|url=http://mpla.math.uoa.gr/~cdimitr/files/publications/AML_33.pdf|journal=Archive for Mathematical Logic|volume=33|issue=4|pages=265–281|doi=10.1007/BF01270626|mr=1294272|url-status=dead|archiveurl=https://web.archive.org/web/20110721083756/http://mpla.math.uoa.gr/~cdimitr/files/publications/AML_33.pdf|archivedate=2011-07-21}}</ref> However, this does not address the question of whether or not the standard proof of PNT can be formalized in PA.

==Computer verifications==
In 2005, Avigad ''et al.'' employed the [[Isabelle theorem prover]] to devise a computer-verified variant of the Erdős–Selberg proof of the PNT.<ref name=Avigad>{{cite journal|first1=Jeremy | last1=Avigad |first2=Kevin |last2=Donnelly |first3=David | last3=Gray |first4=Paul | last4=Raff | title=A formally verified proof of the prime number theorem|year=2008 | mr=2371488 | journal=[[ACM Transactions on Computational Logic]] | volume=9 | issue=1 | pages=2 | doi=10.1145/1297658.1297660 | arxiv=cs/0509025}}</ref> This was the first machine-verified proof of the PNT. Avigad chose to formalize the Erdős–Selberg proof rather than an analytic one because while Isabelle's library at the time could implement the notions of limit, derivative, and [[transcendental function]], it had almost no theory of integration to speak of.<ref name=Avigad/>{{rp|19}}

In 2009, John Harrison employed [[HOL Light]] to formalize a proof employing [[complex analysis]].<ref>{{Cite journal
|title=Formalizing an analytic proof of the Prime Number Theorem
|last=Harrison
|first=John
|url=http://www.cl.cam.ac.uk/~jrh13/papers/mikefest.html
|journal=[[Journal of Automated Reasoning]]
|mr=2544285
|year = 2009 |volume = 43 |pages = 243–261 | issue=3
 |doi=10.1007/s10817-009-9145-6|citeseerx=10.1.1.646.9725
}}</ref> By developing the necessary analytic machinery, including the [[Cauchy integral formula]], Harrison was able to formalize "a direct, modern and elegant proof instead of the more involved 'elementary' Erdős–Selberg<!-- Erdös not Erdős in quote, but Wikipedia editorially corrects it --> argument".

==Prime number theorem for arithmetic progressions==
Let {{math|''π''<sub>''n'',''a''</sub>(''x'')}} denote the number of primes in the [[arithmetic progression]] {{math|''a'', ''a'' + ''n'', ''a'' + 2''n'', ''a'' + 3''n'', ...}} less than {{mvar|x}}. Dirichlet and Legendre conjectured, and de la Vallée Poussin proved, that, if {{mvar|a}} and {{mvar|n}} are [[coprime]], then

:<math>\pi_{n,a}(x) \sim \frac{1}{\varphi(n)} \operatorname{Li}(x),</math>

where {{mvar|φ}} is [[Euler's totient function]]. In other words, the primes are distributed evenly among the residue classes {{math|[''a'']}} [[modular arithmetic|modulo]] {{mvar|n}} with {{math|gcd(''a'', ''n'') {{=}} }}1. This is stronger than [[Dirichlet's theorem on arithmetic progressions]] (which only states that there is an infinity of primes in each class) and can be proved using similar methods used by Newman for his proof of the prime number theorem.<ref>{{cite journal|first=Ivan | last=Soprounov|url=http://academic.csuohio.edu/soprunov_i/pdf/primes.pdf|title=A short proof of the Prime Number Theorem for arithmetic progressions|year=1998}}</ref>

The [[Siegel–Walfisz theorem]] gives a good estimate for the distribution of primes in residue classes.

===Prime number race===
Although we have in particular

:<math>\pi_{4,1}(x) \sim \pi_{4,3}(x), </math>

empirically the primes congruent to 3 are more numerous and are nearly always ahead in this "prime number race"; the first reversal occurs at {{math|''x'' {{=}} 26861}}.<ref name="Granville Martin MAA">
{{cite journal
 | doi = 10.2307/27641834
 | last1 = Granville | first1 = Andrew
 | author1link = Andrew Granville
 | last2 = Martin | first2 = Greg
 | year = 2006
 | title = Prime Number Races
 | journal = [[American Mathematical Monthly]]
 | volume = 113 | issue = 1 | pages = 1–33
 | url = http://www.dms.umontreal.ca/%7Eandrew/PDF/PrimeRace.pdf
 | jstor = 27641834
 | mr = 2202918
}}</ref>{{Rp|1–2}} However Littlewood showed in 1914<ref name="Granville Martin MAA" />{{Rp|2}} that there are infinitely many sign changes for the function

:<math>\pi_{4,1}(x) - \pi_{4,3}(x), </math>

so the lead in the race switches back and forth infinitely many times. The phenomenon that {{math|''π''<sub>4,3</sub>(''x'')}} is ahead most of the time is called [[Chebyshev's bias]]. The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that {{math|''π''(''x'';''a'',''c'')}} and {{math|''π''(''x'';''b'',''c'')}} change places when {{mvar|a}} and {{mvar|b}} are coprime to {{mvar|c}}.<ref name=GuyA4>{{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=A4 }}</ref> [[Andrew Granville|Granville]] and Martin give a thorough exposition and survey.<ref name="Granville Martin MAA" />

==Non-asymptotic bounds on the prime-counting function==
The prime number theorem is an ''asymptotic'' result. It gives an [[Effective results in number theory|ineffective]] bound on {{math|''π''(''x'')}} as a direct consequence of the definition of the limit: for all {{math|''ε'' > 0}}, there is an {{mvar|S}} such that for all {{math|''x'' > ''S''}},
: <math> (1-\varepsilon)\frac {x}{\log x} < \pi(x) < (1+\varepsilon)\frac {x}{\log x}. </math>

However, better bounds on {{math|''π''(''x'')}} are known, for instance [[Pierre Dusart]]'s
:<math> \frac{x}{\log x}\left(1+\frac{1}{\log x}\right) < \pi(x) < \frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2.51}{(\log x)^2}\right). </math>
The first inequality holds for all {{math|''x'' ≥ 599}} and the second one for {{math|''x'' ≥ 355991}}.<ref>{{Cite thesis|last=Dusart|first=Pierre|title=Autour de la fonction qui compte le nombre de nombres premiers|type=PhD thesis|year=1998|url=http://www.unilim.fr/laco/theses/1998/T1998_01.html|language=fr}}</ref>

A weaker but sometimes useful bound for {{math|''x'' ≥ 55}} is<ref>{{cite journal|title=Explicit Bounds for Some Functions of Prime Numbers|first=Barkley|last=Rosser|journal=[[American Journal of Mathematics]]|volume=63|issue=1|year=1941|pages=211–232|doi=10.2307/2371291|jstor=2371291|mr=0003018}}</ref>
:<math> \frac {x}{\log x + 2} < \pi(x) < \frac {x}{\log x - 4}.</math>

In Pierre Dusart's thesis there are stronger versions of this type of inequality that are valid for larger {{mvar|x}}. Later in 2010, Dusart proved:<ref>{{cite arxiv|last=Dusart|first=Pierre|title=Estimates of Some Functions Over Primes without R.H |eprint=1002.0442|class=math.NT|year=2010}}</ref>

:<math>\begin{align}
\frac {x} {\log x - 1} &< \pi(x) &&\text{for } x \ge 5393 , \text{ and}\\
\pi(x) &<  \frac {x} {\log x - 1.1} &&\text{for } x \ge 60184 .
\end{align}</math>

The proof by de la Vallée Poussin implies the following.
For every {{math|''ε'' > 0}}, there is an {{mvar|S}} such that for all {{math|''x'' > ''S''}},
: <math>\frac {x}{\log x - (1-\varepsilon)} < \pi(x) < \frac {x}{\log x - (1+\varepsilon)}.</math>

==Approximations for the ''{{mvar|n}}''th prime number==
As a consequence of the prime number theorem, one gets an asymptotic expression for the {{mvar|n}}th prime number, denoted by {{math|''p''<sub>''n''</sub>}}:
:<math>p_n \sim n \log n.</math>
A better approximation is<ref>{{cite journal|authorlink=Ernesto Cesàro|first=Ernesto|last=Cesàro|year=1894|title=Sur une formule empirique de M. Pervouchine|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|volume=119|pages=848–849|url=http://gallica.bnf.fr/ark:/12148/bpt6k30752|language=fr}}</ref>
:<math> \frac{p_n}{n} = \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} - \frac{(\log\log n)^2 - 6 \log \log n + 11}{2(\log n)^2} + o \left( \frac {1}{(\log n)^2}\right).</math>
Again considering the {{val|2|e=17}}th prime number {{val|8512677386048191063}}, this gives an estimate of {{val|8512681315554715386}}; the first 5 digits match and relative error is about 0.00005%.

[[Rosser's theorem]] states that 
:<math>p_n > n \log n.</math> 
This can be improved by the following pair of bounds:<ref>{{cite journal
| authorlink=J. Barkley Rosser
| last       = Rosser
| first      = Barkley
| date       = 1941
| title      = Explicit bounds for some functions of prime numbers
| jstor        = 2371291
| journal    = American Journal of Mathematics
| volume     = 63
| issue      = 1
| pages      = 211–232
| doi        = 10.2307/2371291
}}</ref>
<ref>{{cite journal|authorlink=Pierre Dusart|last=Dusart|first=Pierre|title=The {{mvar|k}}th prime is greater than {{math|''k''(log ''k'' + log log ''k''−1)}} for {{math|''k'' ≥ 2}}|journal=[[Mathematics of Computation]]|volume=68|issue=225|year=1999|pages=411–415|mr=1620223|doi=10.1090/S0025-5718-99-01037-6}}</ref>
:<math> \log n + \log\log n - 1 < \frac{p_n}{n} < \log n + \log \log n \quad\text{for } n \ge 6. </math>

==Table of {{math|''π''(''x'')}}, {{math|''x'' / log ''x''}}, and {{math|li(''x'')}}==
The table compares exact values of {{math|''π''(''x'')}} to the two approximations {{math|''x'' / log ''x''}} and {{math|li(''x'')}}. The last column, {{math|''x'' / ''π''(''x'')}}, is the average [[prime gap]] below&nbsp;{{mvar|x}}.
:{| class="wikitable" style="text-align: right"
! {{mvar|x}}
! {{math|''π''(''x'')}}
! {{math|''π''(''x'') − {{sfrac|''x''|log ''x''}}}}
! {{math|{{sfrac|''π''(''x'')|''x'' / log ''x''}}}}
! {{math|li(''x'') − ''π''(''x'')}}
! {{math|{{sfrac|''x''|''π''(''x'')}}}}
|-
| 10
| 4
| −0.3
| 0.921
| 2.2
| 2.5{{0|00}}
|-
| 10<sup>2</sup>
| 25
| 3.3
| 1.151
| 5.1
| 4{{0|.000}}
|-
| 10<sup>3</sup>
| 168
| 23{{0|.0}}
| 1.161
| 10{{0|.0}}
| 5.952
|-
| 10<sup>4</sup>
| {{val|1229|fmt=gaps}}
| 143{{0|.0}}
| 1.132
| 17{{0|.0}}
| 8.137
|-
| 10<sup>5</sup>
| {{val|9592|fmt=gaps}}
| 906{{0|.0}}
| 1.104
| 38{{0|.0}}
| 10.425
|-
| 10<sup>6</sup>
| {{val|78498}}
| {{val|6116|fmt=gaps}}{{0|.0}}
| 1.084
| 130{{0|.0}}
| 12.740
|-
| 10<sup>7</sup>
| {{val|664579}}
| {{val|44158}}{{0|.0}}
| 1.071
| 339{{0|.0}}
| 15.047
|-
| 10<sup>8</sup>
| {{val|5761455}}
| {{val|332774}}{{0|.0}}
| 1.061
| 754{{0|.0}}
| 17.357
|-
| 10<sup>9</sup>
| {{val|50847534}}
| {{val|2592592}}{{0|.0}}
| 1.054
| {{val|1701|fmt=gaps}}{{0|.0}}
| 19.667
|-
| 10<sup>10</sup>
| {{val|455052511}}
| {{val|20758029}}{{0|.0}}
| 1.048
| {{val|3104|fmt=gaps}}{{0|.0}}
| 21.975
|-
| 10<sup>11</sup>
| {{val|4118054813}}
| {{val|169923159}}{{0|.0}}
| 1.043
| {{val|11588}}{{0|.0}}
| 24.283
|-
| 10<sup>12</sup>
| {{val|37607912018}}
| {{val|1416705193}}{{0|.0}}
| 1.039
| {{val|38263}}{{0|.0}}
| 26.590
|-
| 10<sup>13</sup>
| {{val|346065536839}}
| {{val|11992858452}}{{0|.0}}
| 1.034
| {{val|108971}}{{0|.0}}
| 28.896
|-
| 10<sup>14</sup>
| {{val|3204941750802}}
| {{val|102838308636}}{{0|.0}}
| 1.033
| {{val|314890}}{{0|.0}}
| 31.202
|-
| 10<sup>15</sup>
| {{val|29844570422669}}
| {{val|891604962452}}{{0|.0}}
| 1.031
| {{val|1052619}}{{0|.0}}
| 33.507
|-
| 10<sup>16</sup>
| {{val|279238341033925}}
| {{val|7804289844393}}{{0|.0}}
| 1.029
| {{val|3214632}}{{0|.0}}
| 35.812
|-
| 10<sup>17</sup>
| {{val|2623557157654233}}
| {{val|68883734693281}}{{0|.0}}
| 1.027
| {{val|7956589}}{{0|.0}}
| 38.116
|-
| 10<sup>18</sup>
| {{val|24739954287740860}}
| {{val|612483070893536}}{{0|.0}}
| 1.025
| {{val|21949555}}{{0|.0}}
| 40.420
|-
| 10<sup>19</sup>
| {{val|234057667276344607}}
| {{val|5481624169369960}}{{0|.0}}
| 1.024
| {{val|99877775}}{{0|.0}}
| 42.725
|-
| 10<sup>20</sup>
| {{val|2220819602560918840}}
| {{val|49347193044659701}}{{0|.0}}
| 1.023
| {{val|222744644}}{{0|.0}}
| 45.028
|-
| 10<sup>21</sup>
| {{val|21127269486018731928}}
| {{val|446579871578168707}}{{0|.0}}
| 1.022
| {{val|597394254}}{{0|.0}}
| 47.332
|-
| 10<sup>22</sup>
| {{val|201467286689315906290}}
| {{val|4060704006019620994}}{{0|.0}}
| 1.021
| {{val|1932355208}}{{0|.0}}
| 49.636
|-
| 10<sup>23</sup>
| {{val|1925320391606803968923}}
| {{val|37083513766578631309}}{{0|.0}}
| 1.020
| {{val|7250186216}}{{0|.0}}
| 51.939
|-
| 10<sup>24</sup>
| {{val|18435599767349200867866}}
| {{val|339996354713708049069}}{{0|.0}}
| 1.019
| {{val|17146907278}}{{0|.0}}
| 54.243
|-
| 10<sup>25</sup>
| {{val|176846309399143769411680}}
| {{val|3128516637843038351228}}{{0|.0}}
| 1.018
| {{val|55160980939}}{{0|.0}}
| 56.546
|-
| [[OEIS]]
| {{OEIS link|id=A006880}}
| {{OEIS link|id=A057835}}
| 
| {{OEIS link|id=A057752}}
| 
|}

The value for {{math|''π''(10<sup>24</sup>)}} was originally computed assuming the [[Riemann hypothesis]];<ref name="Franke">{{cite web |title=Conditional Calculation of {{math|''π''(10<sup>24</sup>)}} |url=http://primes.utm.edu/notes/pi(10%5E24).html |publisher=Chris K. Caldwell |accessdate=2010-08-03}}</ref> it has since been verified unconditionally.<ref name="PlattARXIV2012">{{cite journal | last=Platt | first=David | title=Computing {{math|''π''(''x'')}} analytically | year=2015 | journal=[[Mathematics of Computation]] | volume=84 | issue=293 | pages=1521–1535 | mr=3315519 | arxiv=1203.5712 | doi=10.1090/S0025-5718-2014-02884-6 }}</ref>

==Analogue for irreducible polynomials over a finite field==
There is an analogue of the prime number theorem that describes the "distribution" of [[irreducible polynomial]]s over a [[finite field]]; the form it takes is strikingly similar to the case of the classical prime number theorem.

To state it precisely, let {{math|''F'' {{=}} GF(''q'')}} be the finite field with {{mvar|q}} elements, for some fixed {{mvar|q}}, and let {{mvar|N<sub>n</sub>}} be the number of [[monic polynomial|monic]] ''irreducible'' polynomials over {{mvar|F}} whose [[degree of a polynomial|degree]] is equal to {{mvar|n}}. That is, we are looking at polynomials with coefficients chosen from {{mvar|F}}, which cannot be written as products of polynomials of smaller degree. In this setting, these polynomials play the role of the prime numbers, since all other monic polynomials are built up of products of them. One can then prove that
:<math>N_n \sim \frac{q^n}{n}.</math>
If we make the substitution {{math|''x'' {{=}} ''q''<sup>''n''</sup>}}, then the right hand side is just
:<math>\frac{x}{\log_q x},</math>
which makes the analogy clearer. Since there are precisely {{math|''q''<sup>''n''</sup>}} monic polynomials of degree {{mvar|n}} (including the reducible ones), this can be rephrased as follows: if a monic polynomial of degree {{mvar|n}} is selected randomly, then the probability of it being irreducible is about&nbsp;{{math|{{sfrac|1|''n''}}}}.

One can even prove an analogue of the Riemann hypothesis, namely that
:<math>N_n = \frac{q^n}n + O\left(\frac{q^\frac{n}{2}}{n}\right).</math>

The proofs of these statements are far simpler than in the classical case. It involves a short combinatorial argument,<ref>{{cite journal|last=Chebolu|first=Sunil|first2=Ján|last2=Mináč|title=Counting Irreducible Polynomials over Finite Fields Using the Inclusion {{pi}} Exclusion Principle|journal=Mathematics Magazine|date=December 2011|volume=84|issue=5|pages=369–371|doi=10.4169/math.mag.84.5.369|jstor=10.4169/math.mag.84.5.369|arxiv=1001.0409}}</ref> summarised as follows: every element of the degree {{mvar|n}} extension of {{mvar|F}} is a root of some irreducible polynomial whose degree {{mvar|d}} divides {{mvar|n}}; by counting these roots in two different ways one establishes that
:<math>q^n = \sum_{d\mid n} d N_d,</math>
where the sum is over all [[divisor]]s {{mvar|d}} of {{mvar|n}}. [[Möbius inversion]] then yields
:<math>N_n = \frac{1}{n} \sum_{d\mid n} \mu\left(\frac{n}{d}\right) q^d,</math>
where {{math|''μ''(''k'')}} is the [[Möbius function]]. (This formula was known to Gauss.<!-- although I haven't got a reference for this. -->) The main term occurs for {{math|''d'' {{=}} ''n''}}, and it is not difficult to bound the remaining terms. The "Riemann hypothesis" statement depends on the fact that the largest [[proper divisor]] of {{mvar|n}} can be no larger than {{math|{{sfrac|''n''|2}}}}.

==See also==
* [[Abstract analytic number theory]] for information about generalizations of the theorem.
* [[Landau prime ideal theorem]] for a generalization to prime ideals in algebraic number fields.
* [[Riemann hypothesis]]

==Notes==
{{Reflist|30em}}

==References==
*{{Cite journal |last=Hardy |first=G. H. |last2=Littlewood |first2=J. E. |year=1916 |title=Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes |journal=Acta Mathematica |volume=41 |issue= |pages=119–196 |doi=10.1007/BF02422942 |url=https://zenodo.org/record/2294397/files/article.pdf }}
*{{Cite journal |last=Granville |first=Andrew |year=1995 |title=Harald Cramér and the distribution of prime numbers |journal=Scandinavian Actuarial Journal |volume=1 |issue= |pages=12–28 |url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |issn= |doi=10.1080/03461238.1995.10413946|citeseerx=10.1.1.129.6847 }}

==External links==
* {{springer|title=Distribution of prime numbers|id=p/d033530}}
* [http://www.scs.uiuc.edu/~mainzv/exhibitmath/exhibit/felkel.htm Table of Primes by Anton Felkel].
* [https://www.youtube.com/watch?v=3RfYfMjZ5w0 Short video] visualizing the Prime Number Theorem.
* [http://mathworld.wolfram.com/PrimeFormulas.html Prime formulas] and [http://mathworld.wolfram.com/PrimeNumberTheorem.html Prime number theorem] at [[MathWorld]].
* {{planetmath reference|id=199|title=Prime number theorem}}
* [http://primes.utm.edu/howmany.shtml How Many Primes Are There?] and [http://primes.utm.edu/notes/gaps.html The Gaps between Primes] by Chris Caldwell, [[University of Tennessee at Martin]].
* [http://www.ieeta.pt/~tos/primes.html Tables of prime-counting functions] by Tomás Oliveira e Silva

{{DEFAULTSORT:Prime Number Theorem}}
[[Category:Theorems in analytic number theory]]
[[Category:Theorems about prime numbers]]
[[Category:Logarithms]]