Editing Atle Selberg (section)

==Post-war in Norway==
After the war, he turned to [[sieve theory]], a previously neglected topic which Selberg's work brought into prominence. In a 1947 paper he introduced the [[Selberg sieve]], a method well adapted in particular to providing auxiliary upper bounds, and which contributed to [[Chen's theorem]], among other important results.

In 1948 Selberg submitted two papers in ''[[Annals of Mathematics]]'' in which he proved by elementary means the theorems for [[primes in arithmetic progression]] and the [[prime number theorem|density of primes]].<ref>{{cite journal|jstor=1969455|
 title=An Elementary Proof of the Prime-Number Theorem| url=https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-7230/Selberg-ElemPNT1949.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.math.lsu.edu/~mahlburg/teaching/handouts/2014-7230/Selberg-ElemPNT1949.pdf |archive-date=2022-10-09 |url-status=live| 
 first=Atle|
 last=Selberg|
 journal=Annals of Mathematics|
 date=April 1949|
 pages=305–313|
volume=50|
 issue=2|
doi=10.2307/1969455
|
 s2cid=124153092}}</ref><ref>
{{cite journal|jstor= 1969454| 
 title=An Elementary Proof of Dirichlet's Theorem About Primes in Arithmetic Progression|
 first=Atle|
 last=Selbert|
 journal=Annals of Mathematics|
 date=April 1949|
 pages=297–304|
volume=50| 
 issue=2|
doi=10.2307/1969454 }}</ref> This challenged the widely held view of his time that certain theorems are only obtainable with the advanced methods of [[complex analysis]]. Both results were based on his work on the asymptotic formula
:<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>
where
:<math>\vartheta \left( x \right) = \sum\limits_{p \le x} {\log \left( p \right)}</math>
for primes <math>p</math>. He established this result by elementary means in March 1948, and by July of that year, Selberg and [[Paul Erdős]] each obtained [[elementary proof]]s of the [[prime number theorem]], both using the asymptotic formula above as a starting point.<ref>{{cite journal|author=Spencer, Joel|author2=Graham, Ronald|title=The Elementary Proof of the Prime Number Theorem|journal=The Mathematical Intelligencer|year=2009|volume=31|issue=3|pages=18–23|url=http://www.cs.nyu.edu/spencer/erdosselberg.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.cs.nyu.edu/spencer/erdosselberg.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1007/s00283-009-9063-9|s2cid=15408261|doi-access=free}}</ref> Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter [[priority dispute|dispute]] between the two mathematicians.<ref name=goldfeld>{{Cite journal | last = Goldfeld | first = Dorian | year = 2003 | title = The Elementary Proof of the Prime Number Theorem: an Historical Perspective | journal = Number Theory: New York Seminar | pages = 179–192 |url=http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf |archive-date=2022-10-09 |url-status=live }}</ref><ref name=BaasSkauInterview>{{Cite journal|url=https://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/S0273-0979-08-01223-8.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/S0273-0979-08-01223-8.pdf |archive-date=2022-10-09 |url-status=live |first1=Nils A.|last1= 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|doi-access=free}}</ref>

For his fundamental accomplishments during the 1940s, Selberg received the 1950 [[Fields Medal]].<ref name="FieldsMedals1950">{{cite web | title=Fields Medals 1950 | website=International Mathematical Union (IMU) | url=https://www.mathunion.org/imu-awards/fields-medal/fields-medals-1950 | access-date=2025-03-26}}</ref>