# Atle Selberg

Atle Selberg
Born 14 June 1917
Langesund, Norway
Died 6 August 2007 (aged 90)
Princeton, New Jersey, United States
Nationality Norwegian
Alma mater University of Oslo
Known for Chowla–Selberg formula
Critical line theorem
Maass–Selberg relations
Selberg class
Selberg's conjecture
Selberg integral
Selberg trace formula
Selberg zeta function
Selberg sieve
Spouse(s) Hedvig Liebermann
Awards Abel Prize (honorary) (2002)
Fields Medal (1950)
Wolf Prize (1986)
Gunnerus Medal (2002)
Scientific career
Fields Mathematics
Influences Srinivasa Ramanujan

Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950.

## Early years

Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his brothers also went on to become mathematicians as well, and the remaining one became a professor of engineering. 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. During the war he fought against the German invasion of Norway, and was imprisoned several times. He studied at the University of Oslo and completed his Ph.D. 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 ${\displaystyle \Re (s)={\tfrac {1}{2}}}$.

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.[1][2] 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

${\displaystyle \vartheta \left(x\right)\log \left(x\right)+\sum \limits _{p\leq x}{\log \left(p\right)}\vartheta \left({\frac {x}{p}}\right)=2x\log \left(x\right)+O\left(x\right)}$

where

${\displaystyle \vartheta \left(x\right)=\sum \limits _{p\leq x}{\log \left(p\right)}}$

for primes ${\displaystyle p}$. 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.[3] Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between the two mathematicians.[4][5]

For his fundamental accomplishments during the 1940s, Selberg received the 1950 Fields Medal.

Selberg moved to the United States and settled at the Institute for Advanced Study in Princeton, New Jersey in the 1950s 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 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.[6]

Selberg had two children, Ingrid Selberg and Lars Selberg. Ingrid Selberg is married to playwright Mustapha Matura.

He died at home in Princeton on 6 August 2007 of heart failure.[7]

## References

1. ^ Selberg, Atle (April 1949). "An Elementary Proof of the Prime-Number Theorem" (PDF). Annals of Mathematics. 50: 305–313. doi:10.2307/1969455. JSTOR 1969455.
2. ^ Selbert, Atle (April 1949). "An Elementary Proof of Dirichlet's Theorem About Primes in Arithmetic Progression". Annals of Mathematics. 50: 297–304. doi:10.2307/1969454. JSTOR 1969454.
3. ^ Spencer, Joel; Graham, Ronald (2009). "The Elementary Proof of the Prime Number Theorem" (PDF). The Mathematical Intelligencer. 31 (3): 18–23. doi:10.1007/s00283-009-9063-9.
4. ^ Goldfeld, Dorian (2003). "The Elementary Proof of the Prime Number Theorem: an Historical Perspective". Number Theory: New York Seminar: 179–192.
5. ^ Baas, Nils A.; Skau, Christian F. (2008). "The lord of the numbers, Atle Selberg. On his life and mathematics" (PDF). Bull. Amer. Math. Soc. 45 (4): 617–649. doi:10.1090/S0273-0979-08-01223-8.
6. ^ "Honorary doctors at NTNU". Norwegian University of Science and Technology.
7. ^ "Atle Selberg, 90, Lauded Mathematician, Dies". The New York Times. 17 August 2007.