Atle Selberg
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Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for his work in analytic number theory and the theory of
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s, and in particular for bringing them into relation with
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
. He was awarded the Fields Medal in 1950 and an honorary
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...
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 and
Henrik Henrik is a male given name of Germanic origin, primarily used in Scandinavia, Estonia, Hungary and Slovenia Slovenia ( ; sl, Slovenija ), officially the Republic of Slovenia (Slovene: , abbr.: ''RS''), is a country in Central Europe. It i ...
, were also mathematicians. His other brother, Arne, was a professor of engineering. While he was still at school he was influenced by the work of
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
and he found an exact analytical formula for the partition function as suggested by the works of Ramanujan; however, this result was first published by Hans Rademacher. During the war he fought against the German invasion of Norway, and was imprisoned several times. He studied at the
University of Oslo The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top univers ...
and completed his
PhD PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. albu ...
in 1943.


World War II

During
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
, Selberg worked in isolation due to the
German occupation of Norway The occupation of Norway by Nazi Germany during the Second World War began on 9 April 1940 after Operation Weserübung. Conventional armed resistance to the German invasion ended on 10 June 1940, and Nazi Germany controlled Norway until the ...
. 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 \Re(s)=\tfrac. After the war, he turned to
sieve theory Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed lim ...
, a previously neglected topic which Selberg's work brought into prominence. In a 1947 paper he introduced the
Selberg sieve In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In ...
, 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 The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
'' in which he proved by elementary means the theorems for primes in arithmetic progression and the density of primes. This challenged the widely held view of his time that certain theorems are only obtainable with the advanced methods of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
. Both results were based on his work on the asymptotic formula :\vartheta \left( x \right)\log \left( x \right) + \sum\limits_ \vartheta \left( \right) = 2x\log \left( x \right) + O\left( x \right) where :\vartheta \left( x \right) = \sum\limits_ for primes p. He established this result by elementary means in March 1948, and by July of that year, Selberg and Paul Erdős each obtained
elementary proof In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certa ...
s of the prime number theorem, both using the asymptotic formula above as a starting point. Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between the two mathematicians. 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 Syracuse University (informally 'Cuse or SU) is a Private university, private research university in Syracuse, New York. Established in 1870 with roots in the Methodist Episcopal Church, the university has been nonsectarian since 1920. Locate ...
and later settled at the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in
Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...
in the 1950s where he remained until his death. During the 1950s he worked on introducing
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
into
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, 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 geodesics on a compact Riemann surface and the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is ...
, which is analogous to the duality between the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s and the zeros of the zeta function. He was awarded the 1986
Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...
. He was also awarded an honorary
Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...
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 The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
and the
Gunnerus Medal The Gunnerus Medal ( no, Gunnerusmedaljen) is a medal awarded by the learned society Royal Norwegian Society of Sciences and Letters. It was inaugurated in 1927, and named after Johan Ernst Gunnerus, founder of the learned society. Members of the l ...
. He was elected to the
Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Unive ...
, the
Royal Danish Academy of Sciences and Letters {{Infobox organization , name = The Royal Danish Academy of Sciences and Letters , full_name = , native_name = Det Kongelige Danske Videnskabernes Selskab , native_name_lang = , logo = Royal ...
and the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...
. In 1972 he was awarded an
honorary degree An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or '' ad ho ...
, doctor philos. honoris causa, at the
Norwegian Institute of Technology The Norwegian Institute of Technology (Norwegian: ''Norges tekniske høgskole'', NTH) was a science institute in Trondheim, Norway. It was established in 1910, and existed as an independent technical university for 58 years, after which it was m ...
, later part of
Norwegian University of Science and Technology Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe *Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including th ...
. Selberg had two children, Ingrid Selberg and Lars Selberg. Ingrid Selberg is married to playwright
Mustapha Matura Mustapha Matura (17 December 1939 – 29 October 2019) was a Trinidadian playwright living in London. Characterised by critic Michael Billington as "a pioneering black playwright who opened the doors for his successors", Matura was the first B ...
. He died at home in
Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...
on 6 August 2007 of heart failure.


Selected publications

* ''Atle Selberg Collected Papers: 1'' (Springer-Verlag, Heidelberg), * ''Collected Papers'' (Springer-Verlag, Heidelberg Mai 1998),


References


Further reading

* Albers, Donald J. and Alexanderson, Gerald L. (2011), ''Fascinating Mathematical People: interviews and memoirs'', "Atle Selberg", pp 254–73,
Princeton University Press Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, ...
, . * Interview with Selberg * *


External links

* *
Atle Selberg Archive webpage

Obituary at IAS

Obituary in ''The Times''

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