Atle Selberg

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 ${\displaystyle \mathrm{\Re }(s)=\frac{1}{2}}$.

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. 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

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

where

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

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. 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 Liebermann, who Selberg married in 1947, 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.