View source for Atle Selberg

{{Short description|Norwegian mathematician (1917–2007)}}
{{Use dmy dates|date=April 2017}}
{{More footnotes needed|date=January 2023}}
{{Infobox scientist
| name              = Atle Selberg
| image             = Atle Selberg.jpg
| image_size        =
| birth_date        = {{Birth date|1917|6|14|df=y}}
| birth_place       = [[Langesund]], Norway
| death_date        = {{Death date and age|2007|8|6|1917|6|14|df=y}}
| death_place       = [[Princeton, New Jersey|Princeton]], [[New Jersey]], United States
| residence         = 
| nationality       = Norwegian
| field             = [[Mathematician|Mathematics]]
| workplaces        = {{Plainlist|
*[[Syracuse University]]
*[[Institute for Advanced Study]]<ref name="IAS_profile"/>
}}
| alma_mater        = [[University of Oslo]]
| academic_advisors = 
| known_for         = [[Critical line theorem]] <br /> [[Local rigidity]] <br /> [[Parity problem (sieve theory)|Parity problem]] <br /> [[Weakly symmetric space]] <br />[[Chowla–Selberg formula]] <br />[[Maass–Selberg relations]] <br /> [[Rankin–Selberg method]] <br /> [[Selberg class]] <br /> [[Selberg's conjecture]] <br /> [[Selberg's identity]] <br /> [[Selberg integral]] <br /> [[Selberg trace formula]] <br /> [[Selberg zeta function]] <br /> [[Selberg sieve]]
| awards            = [[Abel Prize]] (honorary) (2002)<br /> [[Fields Medal]] (1950)<br /> [[Wolf Prize in Mathematics|Wolf Prize]] (1986)<br /> [[Gunnerus Medal]] (2002)
| spouse            = Hedvig Liebermann (m. 1947 - died 1995) <br /> Betty Frances ("Mickey") Compton (m. 2003 - 2007) 
}}
'''Atle Selberg''' (14 June 1917 – 6 August 2007) was a Norwegian [[mathematician]] known for his work in [[analytic number theory]] and the theory of [[automorphic form]]s, and in particular for bringing them into relation with [[spectral theory]]. He was awarded the [[Fields Medal]] in 1950 and an honorary [[Abel Prize]] in 2002.

== Early years ==
Selberg was born in [[Langesund]], Norway, the son of teacher Anna Kristina Selberg and mathematician [[Ole Michael Ludvigsen Selberg]]. Two of his three brothers, [[Sigmund Selberg|Sigmund]] and [[Henrik Selberg|Henrik]], were also mathematicians.  His other brother, [[Arne Selberg|Arne]], was a professor of engineering. 
<!-- There seems to be no source for the following anecdote, and the reference given does not mention Selberg:
His first result came at age 14 when he found[[Johann Bernoulli]]'s  remarkable 1697 formula
:<math>\sum_{n=1}^\infty n^{-n} = \int_0^1 x^{-x} dx\quad\quad(=1.291285997...)</math>
which he published as a problem in the book ''Problems and Theorems in Analysis'' (part I, problem 160) by [[Pólya]] and [[Gábor Szegö]].
-->
While he was still at school he was influenced by the work of [[Srinivasa Ramanujan]] and he found an exact analytical formula for the [[Partition function (number theory)|partition function]] as suggested by the works of Ramanujan; however, this result was first published by [[Hans Rademacher]].

He studied at the [[University of Oslo]] and completed his [[PhD|doctorate]] in 1943.

==World War II==
During [[World War II]], Selberg worked in isolation due to the [[German occupation of Norway]]. After the war, his accomplishments became known, including a proof that a positive proportion of the zeros of the [[Riemann zeta function]] lie on the line <math>\Re(s)=\tfrac{1}{2}</math>.

During the war, he fought against the German invasion of Norway, and was imprisoned several times.

==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=interview>{{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]].

== Institute for Advanced Study ==
Selberg moved to the United States and worked as an associate professor at [[Syracuse University]] and later settled at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]] in the 1950s, where he remained until his death.<ref name="IAS_profile">{{cite press release |last1=Ferrara |first1=Christine |title=Atle Selberg 1917–2007 |url=https://www.ias.edu/press-releases/atle-selberg-1917%E2%80%932007 |access-date=14 October 2020 |work=Institute for Advanced Study |date=August 9, 2007 |language=en}}</ref><ref>{{cite news |last1=Maugh II |first1=Thomas H. |date=22 August 2007 |title=Atle Selberg, 90; Researcher 'Left a Profound Imprint on the World of Mathematics' |work=Los Angeles Times |url=https://www.latimes.com/archives/la-xpm-2007-aug-22-me-selberg22-story.html |url-access=subscription |access-date=14 October 2020}}</ref> During the 1950s he worked on introducing [[spectral theory]] into [[number theory]], culminating in his development of the [[Selberg trace formula]], the most famous and influential of his results. In its simplest form, this establishes a duality between the lengths of [[closed geodesic]]s on a [[compact Riemann surface]] and the [[eigenvalues]] of the [[Laplace-Beltrami operator|Laplacian]], which is analogous to the duality between the [[prime number]]s and the zeros of the zeta function.

He generally worked alone. His only coauthor was [[Sarvadaman Chowla]].<ref>{{cite journal|doi=10.1073/pnas.35.7.371 |title=On Epstein's Zeta Function (I) |date=1949 |last1=Chowla |first1=S. |last2=Selberg |first2=A. |journal=Proceedings of the National Academy of Sciences |volume=35 |issue=7 |pages=371–374 |doi-access=free |pmid=16588908 |pmc=1063041 |bibcode=1949PNAS...35..371C }}</ref><ref>{{cite journal|author=Conrey, Brian|author-link=Brian Conrey|title=Math Encounters - Primes and Zeros: A Million-Dollar Mystery|journal=National Museum of Mathematics, YouTube|date=March 12, 2020|url=https://www.youtube.com/watch?v=OS2V6FLFmxU&t=3510s}} (See 58:30 of 1:18:02 in video.)</ref>

Selberg was awarded the 1986 [[Wolf Prize in Mathematics]]. He was also awarded an honorary [[Abel Prize]] in 2002, its founding year, before the awarding of the regular prizes began.

Selberg received many distinctions for his work, in addition to the Fields Medal, the [[Wolf Prize]] and the [[Gunnerus Medal]]. He was elected to the [[Norwegian Academy of Science and Letters]], the [[Royal Danish Academy of Sciences and Letters]] and the [[American Academy of Arts and Sciences]].

In 1972, he was awarded an [[honorary degree]], doctor philos. honoris causa, at the [[Norwegian Institute of Technology]], later part of [[Norwegian University of Science and Technology]].<ref>{{cite web |title=Honorary Doctors |url=http://www.ntnu.edu/phd/honorary-doctors |publisher=Norwegian University of Science and Technology}}</ref>

His first wife, Hedvig, died in 1995. With her, Selberg had two children: Ingrid Selberg (married to playwright [[Mustapha Matura]]) and Lars Selberg. In 2003 Atle Selberg married Betty Frances ("Mickey") Compton (born in 1929).

He died at home in [[Princeton, New Jersey]] on 6 August 2007 of heart failure. Upon his death he was survived by his widow, daughter, son, and four grandchildren.<ref>{{cite news |last=Pearce |first=Jeremy |date=17 August 2007 |title=Atle Selberg, 90, Lauded Mathematician, Dies |work=[[The New York Times]] |url=https://www.nytimes.com/2007/08/17/nyregion/17selberg.html |url-access=limited}}</ref>

== Selected publications ==
{{refbegin}}
* {{cite journal|year=1940|title=Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist|last1=Selberg|first1=Atle|mr=0002626|journal=[[Archiv for Mathematik og Naturvidenskab]]|volume=43|issue=4|zbl=0023.22201|pages=47–50|jfm=66.0377.01}}
* {{cite journal|last1=Selberg|first1=Atle|mr=0010712|year=1942|journal=Skrifter Utgitt av Det Norske Videnskaps-Akademi I Oslo. I. Mat.-Naturv. Klasse|pages=1–59|volume=10|zbl=0028.11101|title=On the zeros of Riemann's zeta-function}}
* {{cite journal|title=On the normal density of primes in small intervals, and the difference between consecutive primes|last1=Selberg|first1=Atle|volume=47|issue=6|pages=87–105|mr=0012624|journal=[[Archiv for Mathematik og Naturvidenskab]]|zbl=0028.34802|year=1943}}
* {{cite journal|title=Bemerkninger om et multiplet integral|last1=Selberg|first1=Atle|year=1944|volume=26|pages=71–78|zbl=0063.06870|journal=Norsk Matematisk Tidsskrift|mr=0018287}}
* {{cite journal|title=Contributions to the theory of the Riemann zeta-function|mr=0020594|last1=Selberg|first1=Atle|year=1946|issue=5|volume=48|pages=89–155|journal=[[Archiv for Mathematik og Naturvidenskab]]|zbl=0061.08402}}
* {{cite journal|mr=0029410|title=An elementary proof of the prime-number theorem|last1=Selberg|first1=Atle|year=1949|journal=[[Annals of Mathematics]]|series=Second Series|volume=50|pages=305–313|zbl=0036.30604|doi=10.2307/1969455|issue=2|jstor=1969455 }}
* {{cite journal|title=Note on a paper by L. G. Sathe|mr=0067143|last1=Selberg|first1=Atle|year=1954|volume=18|pages=83–87|journal=Journal of the Indian Mathematical Society|series=New Series|zbl=0057.28502|issue=1}}
* {{cite journal|mr=0088511|title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series|last1=Selberg|first1=A.|journal=Journal of the Indian Mathematical Society|series=New Series|volume=20|year=1956|pages=47–87|zbl=0072.08201|issue=1–3}}
* {{cite conference|mr=0130324|title=On discontinuous groups in higher-dimensional symmetric spaces|last1=Selberg|first1=Atle|year=1960|pages=147–164|book-title=Contributions to Function Theory|publisher=[[Tata Institute of Fundamental Research]]|location=Bombay|zbl=0201.36603}}
* {{cite conference|mr=0182610|title=On the estimation of Fourier coefficients of modular forms|last1=Selberg|first1=Atle|publisher=[[American Mathematical Society]]|location=Providence, RI|year=1965|pages=1–15|zbl=0142.33903|book-title=Theory of Numbers|series=Proceedings of Symposia in Pure Mathematics|volume=VIII|editor-last1=Whiteman|editor-first1=Albert L.|doi=10.1090/pspum/008/0182610}}
* {{cite journal|mr=0215797|title=On Epstein's zeta-function|last1=Selberg|first1=Atle|last2=Chowla|first2=S.|volume=227|pages=86–110|doi=10.1515/crll.1967.227.86|zbl=0166.05204|journal=[[Journal für die Reine und Angewandte Mathematik]]|year=1967|author-link2=Sarvadaman Chowla}}
* {{cite conference|mr=1220477|last1=Selberg|first1=Atle|title=Old and new conjectures and results about a class of Dirichlet series|pages=367–385|year=1992|book-title=Proceedings of the Amalfi Conference on Analytic Number Theory|editor-last1=Bombieri|editor-first1=E.|editor-last2=Perelli|editor-first2=A.|editor-last3=Salerno|editor-first3=S.|editor-last4=Zannier|editor-first4=U.|publisher=[[University of Salerno|Università di Salerno]]|location=Salerno|zbl=0787.11037|editor-link1=Enrico Bombieri|editor-link4=Umberto Zannier}}
{{refend}}
Selberg's collected works were published in two volumes. The first volume contains 41 articles, and the second volume contains three additional articles, in addition to Selberg's lectures on sieves.
* {{cite encyclopedia|last1=Selberg|first1=Atle|title=Collected Papers. Volume I|publisher=[[Springer-Verlag]]|location=Berlin, Heidelberg|year=1989|isbn=3-540-18389-2|url=https://archive.org/details/atleselbergcolle01selb_531/page/n5/mode/2up|mr=1117906|zbl=0675.10001 }} {{cite book|title=2014 pbk edition|isbn=9783642410215 |last1=Selberg |first1=Atle |date=28 July 2014 |publisher=Springer }}<ref>{{cite web|author=Berg, Michael|date=October 7, 2014|url=https://maa.org/press/maa-reviews/collected-papers-i-atle-selberg |title=review of ''Collected Papers I: Atle Helberg''|website=MAA Reviews, Mathematical Association of America (MAA) }}</ref> [https://mitpressbookstore.mit.edu/book/9783642410215 Description at M.I.T. Press Bookstore]
* {{cite encyclopedia|last1=Selberg|first1=Atle|title=Collected Papers. Volume II|publisher=[[Springer-Verlag]]|location=Berlin, Heidelberg|year=1991|isbn=3-540-50626-8|mr=1295844|zbl=0729.11001 }} [https://mitpressbookstore.mit.edu/book/9783642410222 Description at M.I.T. Press Bookstore]

== References ==
{{reflist}}

==Further reading==
* Albers, Donald J. and [[Gerald L. Alexanderson|Alexanderson, Gerald L.]] (2011), ''Fascinating Mathematical People: interviews and memoirs'', "Atle Selberg", pp 254–73, [[Princeton University Press]], {{isbn|978-0-691-14829-8}}.
* {{Cite journal|url=https://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/
|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}} Interview with Selberg
*{{Cite journal | author = Hejhal, Dennis |date=June–July 2009 | title = Remembering Atle Selberg, 1917–2007 | journal = [[Notices of the American Mathematical Society]] | volume = 56
  | issue = 6 | pages = 692–710 | url = https://www.ams.org/notices/200906/rtx090600692p-corrected.pdf
  |author-link=Dennis Hejhal }}
* {{Cite journal|last=Selberg |url=https://link.springer.com/content/pdf/10.1007/BF02838915.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://link.springer.com/content/pdf/10.1007/BF02838915.pdf |archive-date=2022-10-09 |url-status=live |title=Reflections Around the Ramanujan Centenary|journal=Resonance |year=1996|volume=1 |issue=12 |pages=81–91 |doi=10.1007/BF02838915 |s2cid=120285506 }}

==External links==
{{Commons category|Atle Selberg}}
{{wikiquote}}
* {{MathGenealogy|id=121277}}
* {{MacTutor Biography|id=Selberg}}
* [http://publications.ias.edu/selberg Atle Selberg archive webpage]
* [https://web.archive.org/web/20151122035028/https://www.ias.edu/news/press-releases/2009-30 Obituary at Institute for Advanced Study]
* [https://web.archive.org/web/20110524002831/http://www.timesonline.co.uk/tol/comment/obituaries/article2477242.ece Obituary in ''The Times'']
* [http://arkivportalen.no/side/arkiv/detaljer?arkivId=no-NTNU_arkiv000000008702 Atle Selbergs private archive] exists at NTNU University Library

{{Fields medalists}}
{{Wolf Prize in Mathematics}}

{{Authority control}}

{{DEFAULTSORT:Selberg, Atle}}
[[Category:1917 births]]
[[Category:2007 deaths]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:Fields Medalists]]
[[Category:Institute for Advanced Study faculty]]
[[Category:Members of the Royal Danish Academy of Sciences and Letters]]
[[Category:Members of the Norwegian Academy of Science and Letters]]
[[Category:Norwegian emigrants to the United States]]
[[Category:Norwegian mathematicians]]
[[Category:Number theorists]]
[[Category:People from Bamble]]
[[Category:University of Oslo alumni]]
[[Category:Wolf Prize in Mathematics laureates]]
[[Category:Members of the Royal Swedish Academy of Sciences]]