Atle Selberg

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, were also mathematicians. His other brother, Arne, was a professor of engineering. His first result came at age 14 when he foundJohann Bernoulli's remarkable 1697 formula :\sum_{n=1}^\infty n^{-n} = \int_0^1 x^{-x} dx\quad\quad(=1.291285997...) 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 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 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 \Re(s)=\tfrac{1}{2}.

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 density of primes.{{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}} 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 }} 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 :\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) where :\vartheta \left( x \right) = \sum\limits_{p \le x} {\log \left( p \right)} 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 proofs 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

C.L. Siegel invited Selberg to visit the Institute for Advanced Study in Princeton, New Jersey in 1947. He moved to the United States and was an associate professor at Syracuse University briefly. He returned to the IAS in 1949, where he remained until his death. 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 geodesics on a compact Riemann surface and the eigenvalues of the Laplacian, which is analogous to the duality between the prime numbers and the zeros of the zeta function.

He generally worked alone. His only coauthor was Sarvadaman Chowla.

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.

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.

Selected publications

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.

• Description at M.I.T. Press Bookstore
• Description at M.I.T. Press Bookstore

References

References

1. Spencer, Joel. (2009). "The Elementary Proof of the Prime Number Theorem". The Mathematical Intelligencer.
2. Goldfeld, Dorian. (2003). "The Elementary Proof of the Prime Number Theorem: an Historical Perspective". Number Theory: New York Seminar.
3. (2008). "The lord of the numbers, Atle Selberg. On his life and mathematics". Bull. Amer. Math. Soc..
4. "Fields Medals 1950".
5. (August 9, 2007). "Atle Selberg 1917–2007". Institute for Advanced Study.
6. (22 August 2007). "Atle Selberg, 90; Researcher 'Left a Profound Imprint on the World of Mathematics'". Los Angeles Times.
7. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences.
8. Conrey, Brian. (March 12, 2020). "Math Encounters - Primes and Zeros: A Million-Dollar Mystery". National Museum of Mathematics, YouTube.
9. (2023-03-16). "Atle Selberg portrait on the occasion of receiving the Wolf Foundation Prize in Mathematics".
10. "Honorary Doctors". Norwegian University of Science and Technology.
11. Pearce, Jeremy. (17 August 2007). "Atle Selberg, 90, Lauded Mathematician, Dies". [[The New York Times]].
12. Berg, Michael. (October 7, 2014). "review of ''Collected Papers I: Atle Helberg''".