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| In [[number theory]], the '''prime number theorem''' ('''PNT''') describes the [[asymptotic analysis|asymptotic]] distribution of the [[prime number]]s. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers. It formalizes the intuitive idea that primes become less common as they become larger.
| | My name: Jennifer Lopes<br>My age: 31<br>Country: Switzerland<br>City: Sihlegg <br>Post code: 8834<br>Street: Mattenstrasse 79<br><br>Feel free to visit my web blog [https://twitter.com/thehighsign Saleh Stevens] |
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| Informally speaking, the prime number theorem states that if a [[Randomness|random]] integer is selected in the range of zero to some large integer ''N'', the [[probability]] that the selected integer is prime is about 1 / ln(''N''), where ln(''N'') is the [[natural logarithm]] of ''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 (ln 10<sup>1000</sup> ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (ln 10<sup>2000</sup> ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first ''N'' integers is roughly ln(''N'').<ref>{{cite book|last = Hoffman|first = Paul|title = The Man Who Loved Only Numbers|publisher = Hyperion|year = 1998|page = 227|isbn = 0-7868-8406-1}}</ref>
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| ==Statement of the theorem==
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| [[File:Prime number theorem ratio convergence.svg|thumb|300px|Graph showing ratio of the prime-counting function π(''x'') to two of its approximations, ''x''/ln ''x'' and Li(''x'') (described [[#Prime-counting function in terms of the logarithmic integral|below]]). 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.]]
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| [[File:Prime number theorem absolute error.svg|thumb|300px|Log-log plot showing absolute error of ''x''/ln ''x'' and Li(''x'') (described [[#Prime-counting function in terms of the logarithmic integral|below]]), two approximations to the prime-counting function π(''x''). Unlike the ratios, the differences increase without bound as ''x'' increases.]]
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| <!--[[Image:PrimeNumberTheorem.svg|thumb|right|250px|Graph comparing π(''x'') (red), ''x'' / ln ''x'' (green) and Li(''x'') (blue)]]-->
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| Let π(''x'') be the [[prime-counting function]] that gives 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'' / ln(''x'') is a good approximation to π(''x''), in the sense that the [[limit of a function|limit]] of the ''quotient'' of the two functions π(''x'') and ''x'' / ln(''x'') as ''x'' approaches infinity is 1:
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| : <math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1,</math>
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| known as '''the asymptotic law of distribution of prime numbers'''. Using [[asymptotic notation]] this result can be restated as
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| :<math>\pi(x)\sim\frac{x}{\ln x}.\!</math>
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| This notation (and the [[theorem]]) does ''not'' say anything about the limit of the ''difference'' of the two functions as ''x'' approaches infinity. Instead, the theorem states that ''x''/ln(''x'') approximates π(''x'') in the sense that the [[relative error]] of this approximation approaches 0 as ''x'' approaches infinity.
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| The prime number theorem is equivalent to the statement that the ''n''th prime number ''p''<sub>''n''</sub> is approximately equal to ''n'' ln(''n''), again with the relative error of this approximation approaching 0 as ''n'' approaches infinity. For example, the 200 · 10<sup>15</sup> prime number is 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 (200 · 10<sup>15</sup>)ln(200 · 10<sup>15</sup>) rounds to 7967418752291744388, a relative error of about 6.8%.
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| ==History of the asymptotic law of distribution of prime numbers and its proof==
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| [[Image:Primes - distribution - up to 19 primorial.png|231px|right|thumb|Distribution of primes up to [[:en:Primorial|19#]] (9699690).]]
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| 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 ln(''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: "Im Jahr 1792 oder 1793", according to his own recollection nearly sixty years later in a letter to Encke (1849), he wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter <math>a(=\infty) \frac a{\ln a}</math>". But Gauss never published this conjecture. 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'' / ln(''x'') stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
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| In two papers from 1848 and 1850, the Russian mathematician [[Pafnuty Chebyshev|Pafnuty L'vovich 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 are works of [[Leonhard Euler]], as early as 1737) predating 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 π(''x'')/(''x''/ln(''x'')) as ''x'' goes to infinity exists at all, then it is necessarily equal to one.<ref>{{cite journal |author=N. Costa Pereira |jstor=2322510 |title=A Short Proof of Chebyshev's Theorem |journal=American Mathematical Monthly|year=1985|pages=494–495|volume=92|month=August–September|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 to 1 for all ''x''.<ref>{{cite journal |author=M. Nair |jstor=2320934 |title=On Chebyshev-Type Inequalities for Primes |journal=American Mathematical Monthly |year=1982 |pages=126–129 |volume=89 |month=February |doi=10.2307/2320934 |issue=2}}</ref> 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.
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| 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 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 ζ(''s'') is non-zero for all complex values of the variable ''s'' that have the form ''s'' = 1 + ''it'' with ''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 = 0-521-39789-8}}</ref>
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| 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, and later proofs have 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 |author=D. J. Newman |doi=10.2307/2321853 |jstor=2321853 |issue=9}}</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 |author=D. Zagier |url=http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf |doi=10.2307/2975232 |jstor=2975232 |issue=8}}</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.
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| ==Proof methodology==
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| In a lecture on prime numbers for a general audience, [[Fields medal]]ist [[Terence Tao]] described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the [[von Mangoldt function]]. Then we analyze its notes or frequencies by subjecting it to a process akin to [[Fourier transform]]; this is the [[Mellin transform]]. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs discussed below.<ref>[http://www.youtube.com/watch?v=PtsrAw1LR3E Video] and [http://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf slides] of Tao's lecture on primes, UCLA January 2007.</ref>
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| ==Proof sketch==
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| Here is a sketch of the proof referred to in Tao's lecture mentioned above. 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>\psi(x)</math>, defined by
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| :<math>\psi(x) = \sum_{p^k \le x, \atop p \, \text{is prime}} \log p.</math>
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| This is sometimes written as <math>\psi(x)=\sum_{n\le x} \Lambda(n)</math>, where <math>\Lambda(n)</math> is the [[von Mangoldt function]], namely
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| :<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>
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| It is now relatively easy to check that the PNT is equivalent to the claim that <math>\lim_{x\to\infty} \psi(x)/x=1</math>. Indeed, this follows from the easy estimates
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| :<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>
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| and (using [[big O notation]]) for any ε > 0,
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| :<math>\psi(x) \ge \sum_{x^{1-\epsilon}\le p\le x} \log p\ge\sum_{x^{1-\epsilon}\le p\le x}(1-\epsilon)\log x=(1-\epsilon)(\pi(x)+O(x^{1-\epsilon}))\log x.</math>
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| The next step is to find a useful representation for <math>\psi(x)</math>. Let <math>\zeta(s)</math> be the [[Riemann zeta function]]. It can be shown that <math>\zeta(s)</math> is related to the von Mangoldt function <math>\Lambda(n)</math>, and hence to <math>\psi(x)</math>, via the relation
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| :<math>-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n = 1}^\infty \Lambda(n) n^{-s}. </math>
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| 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
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| :<math>\psi(x) = x - \sum_{\rho} \frac{x^\rho}{\rho} - \log(2\pi)</math>
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| holds, where the sum is over all zeros (trivial and non-trivial) 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 ''x'' (claimed to be the correct asymptotic order of <math>\psi(x)</math>) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.
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| 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:
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| :<math>\sum_{n=1}^\infty \frac{1}{2n\,x^{2n}} = -\frac{1}{2}\ln\left(1-\frac{1}{x^2}\right),</math>
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| which vanishes for a large ''x''. The nontrivial zeros, namely those on the critical strip <math>0\le \Re(s)\le 1</math>, can potentially be of an asymptotic order comparable to the main term ''x'' if <math>\Re(\rho)=1</math>, so we need to show that all zeros have real part strictly less than 1.
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| To do this, we take for granted that <math>\zeta(s)</math> is meromorphic in the half-plane <math>\Re(s)>0</math>, and is analytic there except for a simple pole at <math>s=1</math>, and that there is a product formula <math>\zeta(s)=\prod_p(1-p^{-s})^{-1} </math> for <math>\Re(s)>1.</math> This product formula follows from the existence of unique prime factorization of integers, and shows that <math>\zeta(s)</math> is never zero in this region, so that its logarithm is defined there and <math>\log\zeta(s)=-\sum_p\log (1-p^{-s} )=\sum_{p,n}p^{-ns}/n.</math> Write <math>s=x+iy</math>; then
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| :<math>|\zeta(x+iy)|=\exp(\sum_{n,p}\frac{\cos ny\log p}{np^{nx}}).</math>
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| Now observe the identity <math>3+4\cos\phi+\cos2\phi=2(1+\cos\phi)^2\ge 0,</math> so that
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| :<math>|\zeta(x)^3\zeta(x+iy)^4\zeta(x+2iy)|=\exp\sum_{n,p}\frac{3+4\cos(ny\log p) +\cos (2ny\log p)}{np^{nx}}\ge 1</math>
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| for all <math>x> 1</math>. Suppose now that <math>\zeta(1+iy)=0</math>. Certainly <math>y</math> is not zero, since <math>\zeta(s)</math> has a simple pole at <math>s=1</math>. Suppose that <math>x>1</math> and let <math>x</math> tend to <math>1</math> from above. Since <math>\zeta(s)</math> has a simple pole at <math>s=1</math> and <math>\zeta(x+2iy)</math> stays analytic, the left hand side in the previous inequality tends to <math>0</math>, a contradiction.
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| Finally, we can conclude that the PNT is "morally" 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>\psi(x)</math> does not converge absolutely but only conditionally and in a "principal value" sense. There are several ways around this problem but all of them require rather delicate complex-analytic estimates that are beyond the scope of this article. 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 = 0-486-41740-9}}</ref> provides the details.
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| ==Prime-counting function in terms of the logarithmic integral==
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| In a handwritten note on a reprint of his 1838 paper "Sur l'usage des séries infinies dans la théorie des nombres", which he mailed to [[Carl Friedrich Gauss]], [[Peter Gustav Lejeune Dirichlet]] conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to π(''x'') is given by the [[logarithmic integral function|offset logarithmic integral]] function Li(''x''), defined by
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| :<math> \mathrm{Li}(x) = \int_2^x \frac1{\ln t} \,\mathrm{d}t = \mathrm{li}(x) - \mathrm{li}(2). </math>
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| Indeed, this integral is strongly suggestive of the notion that the 'density' of primes around ''t'' should be 1/ln''t''. This function is related to the logarithm by the [[asymptotic expansion]]
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| :<math> \mathrm{Li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k}
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| = \frac{x}{\ln x} + \frac{x}{(\ln x)^2} + \frac{2x}{(\ln x)^3} + \cdots. </math>
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| So, the prime number theorem can also be written as π(''x'') ~ Li(''x''). In fact, it follows from the proof of Hadamard and de la Vallée Poussin that
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| :<math> \pi(x)={\rm Li} (x) + O \left(x \mathrm{e}^{-a\sqrt{\ln x}}\right) \quad\text{as } x \to \infty</math>
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| for some positive constant ''a'', where ''O''(…) is the [[big O notation]]. This has been improved to
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| :<math>\pi(x)={\rm Li} (x) + O \left(x \, \exp \left( -\frac{A(\ln x)^{3/5}}{(\ln \ln x)^{1/5}} \right) \right).</math>
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| Because of the connection between the Riemann zeta function and 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|date=December 1901|pages=159–182|author=Helge von Koch|doi=10.1007/BF02403071}} {{fr icon}}</ref> that, [[if and only if]] the Riemann hypothesis is true, the error term in the above relation can be improved to
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| :<math> \pi(x) = {\rm Li} (x) + O\left(\sqrt x \ln x\right). </math>
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| The constant involved in the big O notation was estimated in 1976 by [[Lowell Schoenfeld]]:<ref>{{Cite journal |last=Schoenfeld |first=Lowell |title=Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x). II |journal=Mathematics of Computation |volume=30 |issue=134 |year=1976 |pages=337–360 |doi=10.2307/2005976 |jstor=2005976 }}.</ref> assuming the Riemann hypothesis,
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| :<math>|\pi(x)-{\rm li}(x)|<\frac{\sqrt x\,\ln x}{8\pi}</math>
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| for all ''x'' ≥ 2657. He also derived a similar bound for the [[Chebyshev function|Chebyshev prime-counting function]] ψ:
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| :<math>|\psi(x)-x|<\frac{\sqrt x\,\ln^2 x}{8\pi}</math>
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| for all ''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|1/''f'' noise]] and to also correspond to the [[Tweedie distributions|Tweedie compound Poisson distribution]].<ref>{{cite journal|last=Kendal|first=WS|title=Fluctuation scaling and 1/f noise: shared origins from the Tweedie family of statistical distributions|journal=J Basic Appl Phys|year=2013|volume=2|pages=40–49}}</ref> 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, B|coauthors=Martinez, JR & Tsao, M|title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics|year=1994|volume=21|pages=223–243}}</ref>
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| The logarithmic integral Li(''x'') is larger than π(''x'') for "small" values of ''x''. This is because it is (in some sense) counting not primes, but prime powers, where a power ''p''<sup>''n''</sup> of a prime ''p'' is counted as 1/''n'' of a prime. This suggests that Li(''x'') should usually be larger than π(''x'') by roughly Li(''x''<sup>1/2</sup>)/2, and in particular should usually be larger than π(''x''). However, in 1914, [[John Edensor Littlewood|J. E. Littlewood]] proved that this is not always the case. The first value of ''x'' where π(''x'') exceeds Li(''x'') is probably around ''x'' = 10<sup>316</sup>; see the article on [[Skewes' number]] for more details.
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| ==Elementary proofs==
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| 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">D. Goldfeld [http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf The elementary proof of the prime number theorem: an historical perspective].</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. There is no rigorous and widely accepted definition of the notion of [[elementary proof]] in number theory. One definition is "a proof 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.
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| In March 1948, [[Atle Selberg]] established, by elementary means, the asymptotic formula
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| :<math>\vartheta \left( x \right)\log \left( x \right) + \sum\limits_{p \le x} {\log \left( p \right)}\ \vartheta \left( {\frac{x}{p}} \right) = 2x\log \left( x \right) + O\left( x \right)</math> | |
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| where
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| :<math>\vartheta \left( x \right) = \sum\limits_{p \le x} {\log \left( p \right)}</math>
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| for primes <math>p</math>. 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|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> These proofs effectively laid to rest the notion that the PNT was "deep," and showed that technically "elementary" methods (in other words Peano arithmetic) were more powerful than had been believed to be the case. In 1994, Charalambos Cornaros and Costas Dimitracopoulos proved the PNT using only <math>I\Delta_0+\exp</math>,<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}}</ref> a formal system far weaker than Peano arithmetic. On the history of the elementary proofs of the PNT, including the Erdős–Selberg [[priority dispute]], see [[Dorian Goldfeld]].<ref name="Goldfeld Historical Perspective" />
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| ==Computer verifications==
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| 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>{{cite arXiv|author=Jeremy Avigad, Kevin Donnelly, David Gray, Paul Raff|eprint=cs.AI/0509025|title=A formally verified proof of the prime number theorem|class=cs.AI|year=2005}}</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 (Avigad et al. p. 19).
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| In 2009, John Harrison employed [[HOL Light]] to formalize a proof employing [[complex analysis]].<ref>{{Cite journal
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| |title=Formalizing an analytic proof of the Prime Number Theorem. (Dedicated to Mike Gordon on the occasion of his 60th birthday)
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| |url=http://www.cl.cam.ac.uk/~jrh13/papers/mikefest.html
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| |journal=Journal of Automated Reasoning
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| |year = 2009, volume = 43, pages = 243––261}}</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’ Erdös–Selberg<!-- sic: ö not ő in quote--> argument.”
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| ==Prime number theorem for arithmetic progressions==
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| Let <math>\pi_{n,a}(x)</math> denote the number of primes in the [[arithmetic progression]] ''a'', ''a'' + ''n'', ''a'' + 2''n'', ''a'' + 3''n'', … less than ''x''. [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] and [[Adrien-Marie Legendre|Legendre]] conjectured, and [[Charles Jean de la Vallée-Poussin|Vallée-Poussin]] proved, that, if ''a'' and ''n'' are [[coprime]], then
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| :<math> | |
| \pi_{n,a}(x) \sim \frac{1}{\phi(n)}\mathrm{Li}(x),
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| </math>
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| where φ(·) is the [[Euler's totient function]]. In other words, the primes are distributed evenly among the residue classes [''a''] [[modular arithmetic|modulo]] ''n'' with gcd(''a'', ''n'') = 1. This can be proved using similar methods used by Newman for his proof of the prime number theorem.<ref>{{cite journal|author=Ivan 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>
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| The [[Siegel–Walfisz theorem]] gives a good estimate for the distribution of primes in residue classes.
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| | |
| ===Prime number race===
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| 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 ''x'' = 26,861.<ref name="Granville Martin MAA">
| |
| {{cite journal
| |
| | doi = 10.2307/27641834
| |
| | last1 = Granville | first1 = Andrew
| |
| | last2 = Martin | first2 = Greg
| |
| | authorlink = Andrew Granville
| |
| |date=January 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
| |
| }}</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 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 π(''x'';''a'',''c'') and π(''x'';''b'',''c'') change places when ''a'' and ''b'' are coprime to ''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> Granville and Martin give a thorough exposition and survey.<ref name="Granville Martin MAA" />
| |
| | |
| ==Bounds on the prime-counting function==
| |
| The prime number theorem is an ''asymptotic'' result. Hence, it cannot be used to ''bound'' π(''x'').
| |
| | |
| However, some bounds on π(''x'') are known, for instance [[Pierre Dusart]]'s
| |
| :<math> \frac{x}{\ln x}\left(1+\frac{1}{\ln x}\right) < \pi(x) < \frac{x}{\ln x}\left(1+\frac{1}{\ln x}+\frac{2.51}{(\ln x)^2}\right). </math>
| |
| The first inequality holds for all ''x'' ≥ 599 and the second one for ''x'' ≥ 355991.<ref>{{Cite book|last=Dusart|first=Pierre|title=Autour de la fonction qui compte le nombre de nombres premiers|work=PhD Thesis|year=1998|url=http://www.unilim.fr/laco/theses/1998/T1998_01.html}} {{fr icon}}</ref>
| |
| | |
| A weaker but sometimes useful bound is
| |
| :<math> \frac {x}{\ln x + 2} < \pi(x) < \frac {x}{\ln x - 4}</math>
| |
| for ''x'' ≥ 55.<ref>{{cite journal|title=Explicit Bounds for Some Functions of Prime Numbers|author=Barkley Rosser|journal=American Journal of Mathematics|volume=63|issue=1|date=January 1941|pages=211–232|doi=10.2307/2371291|jstor=2371291}}</ref> In Dusart's thesis there are stronger versions of this type of inequality that are valid for larger ''x''.
| |
| | |
| The proof by de la Vallée-Poussin implies the following.
| |
| For every ε > 0, there is an ''S'' such that for all ''x'' > ''S'',
| |
| : <math>\frac {x}{\ln x - (1-\varepsilon)} < \pi(x) < \frac {x}{\ln x - (1+\varepsilon)}.</math>
| |
| | |
| ==Approximations for the ''n''th prime number==
| |
| As a consequence of the prime number theorem, one gets an [[Asymptotic analysis|asymptotic]] expression for the ''n''th prime number, denoted by ''p''<sub>''n''</sub>:
| |
| :<math>p_n \sim n \ln n.</math>
| |
| A better approximation is
| |
| :<math>{ \frac{p_n}{n} = \ln n + \ln \ln n - 1 + \frac{\ln \ln n - 2}{\ln n} - \frac{(\ln\ln n)^2 - 6 \ln \ln n + 11}{2(\ln n)^2} + o \left( \frac {1}{(\ln n)^2}\right).}</math><ref>{{cite journal|author=[[Ernesto Cesàro|Ernest 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}} {{fr icon}}</ref>
| |
| Again considering the 200 · 10<sup>15</sup> prime number 8512677386048191063, this gives an estimate of 8512681315554715386; the first 5 digits match and relative error is about 0.00005%.
| |
| | |
| [[Rosser's theorem]] states that ''p''<sub>''n''</sub> is larger than ''n'' ln ''n''. This can be improved by the following pair of bounds:<ref>{{cite book|author=[[Eric Bach]], [[Jeffrey Shallit]]|title=Algorithmic Number Theory|volume=1|year=1996|publisher=MIT Press|isbn=0-262-02405-5|page=233}}</ref><ref>{{cite journal|author=[[Pierre Dusart]]|url=http://www.ams.org/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf|title=The ''k''th prime is greater than ''k''(ln ''k'' + ln ln ''k''−1) for ''k'' ≥ 2|journal=Mathematics of Computation|volume=68|year=1999|pages=411–415}}</ref>
| |
| :<math> \ln n + \ln\ln n - 1 < \frac{p_n}{n} < \ln n + \ln \ln n \quad\text{for } n \ge 6. </math>
| |
| | |
| ==Table of π(''x''), ''x'' / ln ''x'', and li(''x'')==
| |
| The table compares exact values of π(''x'') to the two approximations ''x'' / ln ''x'' and li(''x''). The last column, ''x'' / π(''x''), is the average [[prime gap]] below ''x''.
| |
| :{| class="wikitable" style="text-align: right"
| |
| ! ''x''
| |
| ! π(''x'')
| |
| ! π(''x'') − ''x'' / ln ''x''
| |
| ! π(''x'') / (''x'' / ln ''x'')
| |
| ! li(''x'') − π(''x'')
| |
| ! ''x'' / π(''x'')
| |
| |-
| |
| | 10
| |
| | 4
| |
| | −0.3
| |
| | 0.921
| |
| | 2.2
| |
| | 2.500
| |
| |-
| |
| | 10<sup>2</sup>
| |
| | 25
| |
| | 3.3
| |
| | 1.151
| |
| | 5.1
| |
| | 4.000
| |
| |-
| |
| | 10<sup>3</sup>
| |
| | 168
| |
| | 23
| |
| | 1.161
| |
| | 10
| |
| | 5.952
| |
| |-
| |
| | 10<sup>4</sup>
| |
| | 1,229
| |
| | 143
| |
| | 1.132
| |
| | 17
| |
| | 8.137
| |
| |-
| |
| | 10<sup>5</sup>
| |
| | 9,592
| |
| | 906
| |
| | 1.104
| |
| | 38
| |
| | 10.425
| |
| |-
| |
| | 10<sup>6</sup>
| |
| | 78,498
| |
| | 6,116
| |
| | 1.084
| |
| | 130
| |
| | 12.740
| |
| |-
| |
| | 10<sup>7</sup>
| |
| | 664,579
| |
| | 44,158
| |
| | 1.071
| |
| | 339
| |
| | 15.047
| |
| |-
| |
| | 10<sup>8</sup>
| |
| | 5,761,455
| |
| | 332,774
| |
| | 1.061
| |
| | 754
| |
| | 17.357
| |
| |-
| |
| | 10<sup>9</sup>
| |
| | 50,847,534
| |
| | 2,592,592
| |
| | 1.054
| |
| | 1,701
| |
| | 19.667
| |
| |-
| |
| | 10<sup>10</sup>
| |
| | 455,052,511
| |
| | 20,758,029
| |
| | 1.048
| |
| | 3,104
| |
| | 21.975
| |
| |-
| |
| | 10<sup>11</sup>
| |
| | 4,118,054,813
| |
| | 169,923,159
| |
| | 1.043
| |
| | 11,588
| |
| | 24.283
| |
| |-
| |
| | 10<sup>12</sup>
| |
| | 37,607,912,018
| |
| | 1,416,705,193
| |
| | 1.039
| |
| | 38,263
| |
| | 26.590
| |
| |-
| |
| | 10<sup>13</sup>
| |
| | 346,065,536,839
| |
| | 11,992,858,452
| |
| | 1.034
| |
| | 108,971
| |
| | 28.896
| |
| |-
| |
| | 10<sup>14</sup>
| |
| | 3,204,941,750,802
| |
| | 102,838,308,636
| |
| | 1.033
| |
| | 314,890
| |
| | 31.202
| |
| |-
| |
| | 10<sup>15</sup>
| |
| | 29,844,570,422,669
| |
| | 891,604,962,452
| |
| | 1.031
| |
| | 1,052,619
| |
| | 33.507
| |
| |-
| |
| | 10<sup>16</sup>
| |
| | 279,238,341,033,925
| |
| | 7,804,289,844,393
| |
| | 1.029
| |
| | 3,214,632
| |
| | 35.812
| |
| |-
| |
| | 10<sup>17</sup>
| |
| | 2,623,557,157,654,233
| |
| | 68,883,734,693,281
| |
| | 1.027
| |
| | 7,956,589
| |
| | 38.116
| |
| |-
| |
| | 10<sup>18</sup>
| |
| | 24,739,954,287,740,860
| |
| | 612,483,070,893,536
| |
| | 1.025
| |
| | 21,949,555
| |
| | 40.420
| |
| |-
| |
| | 10<sup>19</sup>
| |
| | 234,057,667,276,344,607
| |
| | 5,481,624,169,369,960
| |
| | 1.024
| |
| | 99,877,775
| |
| | 42.725
| |
| |-
| |
| | 10<sup>20</sup>
| |
| | 2,220,819,602,560,918,840
| |
| | 49,347,193,044,659,701
| |
| | 1.023
| |
| | 222,744,644
| |
| | 45.028
| |
| |-
| |
| | 10<sup>21</sup>
| |
| | 21,127,269,486,018,731,928
| |
| | 446,579,871,578,168,707
| |
| | 1.022
| |
| | 597,394,254
| |
| | 47.332
| |
| |-
| |
| | 10<sup>22</sup>
| |
| | 201,467,286,689,315,906,290
| |
| | 4,060,704,006,019,620,994
| |
| | 1.021
| |
| | 1,932,355,208
| |
| | 49.636
| |
| |-
| |
| | 10<sup>23</sup>
| |
| | 1,925,320,391,606,803,968,923
| |
| | 37,083,513,766,578,631,309
| |
| | 1.020
| |
| | 7,250,186,216
| |
| | 51.939
| |
| |-
| |
| | 10<sup>24</sup>
| |
| | 18,435,599,767,349,200,867,866
| |
| | 339,996,354,713,708,049,069
| |
| | 1.019
| |
| | 17,146,907,278
| |
| | 54.243
| |
| |-
| |
| | 10<sup>25</sup>
| |
| | 176,846,309,399,143,769,411,680
| |
| | 3,128,516,637,843,038,351,228
| |
| | 1.018
| |
| | 55,160,980,939
| |
| | 56.546
| |
| |-
| |
| | [[OEIS]]
| |
| | {{OEIS link|id=A006880}}
| |
| | {{OEIS link|id=A057835}}
| |
| |
| |
| | {{OEIS link|id=A057752}}
| |
| |
| |
| |}
| |
| | |
| The value for π(10<sup>24</sup>) was originally computed assuming the [[Riemann hypothesis]];<ref name="Franke">{{cite web |title=Conditional Calculation of pi(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 web |title=Computing π(x) Analytically) |url=http://arxiv.org/abs/1203.5712 |accessdate=Jul 25, 2012}}</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 ''F'' = GF(''q'') be the finite field with ''q'' elements, for some fixed ''q'', and let ''N''<sub>''n''</sub> be the number of [[monic polynomial|monic]] ''irreducible'' polynomials over ''F'' whose [[degree of a polynomial|degree]] is equal to ''n''. That is, we are looking at polynomials with coefficients chosen from ''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 ''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 ''q''<sup>''n''</sup> monic polynomials of degree ''n'' (including the reducible ones), this can be rephrased as follows: if a monic polynomial of degree ''n'' is selected randomly, then the probability of it being irreducible is about 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^{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|coauthors=Ján Mináč|title=Counting Irreducible Polynomials over Finite Fields Using the Inclusion-Exclusion Principle|journal=Mathematics Magazine|date=December 2011|volume=84|issue=5|pages=369–371|doi=10.4169/math.mag.84.5.369|url=http://www.jstor.org/stable/10.4169/math.mag.84.5.369}}</ref> summarised as follows. Every element of the degree ''n'' extension of ''F'' is a root of some irreducible polynomial whose degree ''d'' divides ''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 ''d'' of ''n''. [[Möbius inversion]] then yields
| |
| :<math>N_n = \frac1n \sum_{d\mid n} \mu(n/d) q^d,</math> | |
| where 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 ''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 ''n'' can be no larger than ''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|colwidth=30em}}
| |
| | |
| ==References==
| |
| *{{Cite journal |last=Hardy |first=G. H. |last2=Littlewood |first2=J. E. |lastauthoramp=yes |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 }}
| |
| *{{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= }}
| |
| | |
| ==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].
| |
| * [http://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
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| {{DEFAULTSORT:Prime Number Theorem}}
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| [[Category:Theorems in analytic number theory]]
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| [[Category:Theorems about prime numbers]]
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| [[Category:Logarithms]]
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| {{Link FA|sl}}
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| {{Link FA|vo}}
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| [[pl:Liczba pierwsza#Twierdzenie o liczbach pierwszych]]
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