Editing Prime number theorem (section)

== 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>{{citation|last=Gauss|first=C. F.|author-link=Carl Friedrich Gauss|year=1863|title=Werke|publisher=Teubner|location=Göttingen|edition=1st|volume=2|pages=444–447|url=https://archive.org/details/carlfriedrichgu00gausgoog/page/444/mode/2up}}.</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 as {{mvar|x}} goes to infinity of {{math|''π''(''x'') / (''x'' / log(''x''))}}  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 0.92129 and 1.10555, 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><ref name="Goldfeld Historical Perspective" /> 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, 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 {{math|''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]]<ref name="Hadamard1896" /> and [[Charles Jean de la Vallée Poussin]]<ref name="de la Vallée Poussin1896" /> 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 nonzero 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]]<ref name="Selberg1949" /> and [[Paul Erdős]]<ref name="Erdős1949">{{citation|last=Erdős|first=Paul|author-link=Paul Erdős|date=1949-07-01|title=On a new method in elementary number theory which leads to an elementary proof of the prime number theorem|journal=Proceedings of the National Academy of Sciences|publisher=National Academy of Sciences|location=U.S.A.|volume=35|issue=7|pages=374–384|doi=10.1073/pnas.35.7.374|pmid=16588909 |pmc=1063042 |bibcode=1949PNAS...35..374E |url=https://www.renyi.hu/~p_erdos/1949-02.pdf|doi-access=free }}</ref> (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]].<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 name=":0">{{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 [[Elementary proof|non-elementary]] in the sense that it uses [[Cauchy's integral theorem]] from complex analysis.