# Atle Selberg

### Times obituary

Atle Selberg 'Mathematician's mathematician' whose work on prime number theory influenced many branches of his discipline

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Atle Selberg was one of the greatest mathematicians of the 20th century. He was a member of the faculty of the Institute for Advanced Study in Princeton, New Jersey from 1951 to 1987.

His most famous early work concerned his elementary proof of the Prime Number Theorem in 1949, in the Annals of Mathematics. Later, in 1956, he formulated the Selberg Trace Formula, combining number theory with the geometry of surfaces. The ramifications of this work became Selberg's main focus after 1956, and have proved very influential in a number of areas of mathematics.

Atle Selberg was born in Langesund, Norway, in 1917, the youngest of nine children of an academic family. He was interested in mathematics from boyhood, and was particularly fascinated by the life and work of the great Indian mathematician Srinivasa Ramanujan (1887-1920), who worked on number theory.

Selberg studied at the University of Oslo, where he took his doctorate in 1943. He held a research fellowship there in 1942-47. In 1947 he married, and went to America, where he remained for the rest of his life.

He spent 1947-48 at the Institute for Advanced Study in Princeton, New Jersey, then 1948-49 at Syracuse University, returning to Princeton in 1949 as a permanent member of staff. He became a professor there in 1951, retiring in 1987.

Selberg was awarded the Fields Medal (roughly the equivalent in mathematics to a Nobel prize, but restricted to those under 40) at the 1950 International Congress of Mathematicians at Harvard.

From 1949 he concentrated on elementary proof of the Prime Number Theorem. The most fundamental mathematical objects are the positive integers, or natural numbers, such as 1, 2, 3 and so on. Each such number can be factorised into a product of primes -- numbers such as 2, 3, 5, 7, 11, 13, 17, 19 etc -- that cannot be factorised further.

It had been known since ancient times (by Euclid, for example, around 300BC) that there are infinitely many such primes. It was noticed that the primes became sparser as one proceeded to higher numbers, and the problem was to estimate how many primes there are up to some large level.

Carl Friedrich Gauss conjectured in the early 19th century (from numerical evidence) the number of primes up to a large level. This conjecture was proved, as the Prime Number Theorem, in 1896, simultaneously by Jacques Hadamard and by Charles-Jean de la Vallée Poussin.

However, their proofs involved complex numbers (pairs of the form$x + iy$, where $x$ and $y$ are real numbers, and $i$ denotes the square root of minus one; thus any product of $i$ with itself is replaced by minus one).

Since integers are more basic than real numbers, this use of complex numbers -- less basic still -- to solve the outstanding problem about integers and primes was felt by the mathematical community to be a blemish from the aesthetic point of view.

The search was on for an "elementary" proof of the Prime Number Theorem, one using real numbers only. In an echo of the earlier coincidence of 1896, in 1949 elementary proofs were provided by two people: Selberg and the Hungarian mathematician, Paul Erdös (1913-96). There were plans at one time for their papers to appear consecutively, but in the end Selberg published his first, in two papers in the Annals of Mathematics in 1949.

The greatest work in prime number theory between the conjecture of the Prime Number Theorem and its proof was that of the German mathematician Bernhard Riemann in the mid-19th century.

Perhaps even more influential was Selberg's work of 1956 on the Selberg Trace Formula. This combined ideas of number theory with the geometry of surfaces. It became the main focus of his later work, and has had great influence on the development of a number of branches of mathematics. Selberg remains one of the greatest mathematicians of the 20th century. He was hard working, continuing to be mathematically active well into his eighties. He was also modest, saying of his own work that it was usually based on rather simple ideas. To his colleagues, he was a mathematician of extraordinary depth and power, well described by one colleague as "a mathematician's mathematician".

Selberg was widely honoured. He was awarded an honorary doctorate by the University of Trondheim (1972), the Wolf Prize in Mathematics (1986), membership of the Royal Norwegian Academy of Sciences and Letters, the Royal Danish Academy of Sciences and Letters, the Royal Swedish Academy of Sciences, the American Academy of Arts and Sciences, the Indian National Science Academy and the London Mathematical Society.

An international symposium was held to honour his 70th birthday. His collected works were published in two volumes (Springer, 1989 and 1991).

Selberg married Hedvig Liebermann in 1947; they had a daughter and a son. Hedvig also worked at the Institute for Advanced Study. She predeceased him in 1995.

Selberg is survived by his second wife, Betty Compton Selberg, his two children and two stepdaughters.

Atle Selberg, mathematician, was born on June 14, 1917. He died on August 6, 2007, aged 90