Editing Prime number theorem (section)

{{short description|Characterization of how many integers are prime}}
{{log(x)}}
{{Duplication|dupe=Prime-counting function|discuss=Talk:Prime number theorem#Too much duplication in Prime number theorem and Prime-counting function|date=December 2024}}
In [[mathematics]], 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]]<ref name="Hadamard1896">{{citation|last=Hadamard|first=Jacques|author-link=Jacques Hadamard|year=1896|title=Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques.|journal=Bulletin de la Société Mathématique de France|publisher=Société Mathématique de France|volume=24|pages=199–220|url=http://www.numdam.org/item/?id=BSMF_1896__24__199_1 |archive-url=https://web.archive.org/web/20240910153636/http://www.numdam.org/item/?id=BSMF_1896__24__199_1 |archive-date=2024-09-10 }}</ref> and [[Charles Jean de la Vallée Poussin]]<ref name="de la Vallée Poussin1896">{{citation|last=de la Vallée Poussin|first=Charles-Jean|author-link=Charles Jean de la Vallée Poussin|year=1896|title=Recherches analytiques sur la théorie des nombres premiers.|journal=Annales de la Société scientifique de Bruxelles|publisher=Imprimeur de l'Académie Royale de Belgique|volume=20 B; 21 B|pages=183-256, 281-352, 363-397; 351-368|url=http://sciences.amisbnf.org/fr/livre/recherches-analytiques-de-la-theorie-des-nombres-premiers}}</ref> 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]] (the number of primes less than or equal to ''N'') 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/page/227|url-access = registration|publisher = Hyperion Books|year = 1998|page = [https://archive.org/details/manwholovedonlyn00hoff/page/227 227]|isbn = 978-0-7868-8406-3|mr = 1666054|location = New York}}</ref>