Atle Selberg

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Atle Selberg (born June 17, 1917) is 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. Selberg was born in Langesund, Norway. His first result came at age 14 when he discovered the remarkable formula

$\sum_{n=1}^\infty n^{-n} = \int_0^1 x^{-x} dx$

which he published as a problem in the book Problems and Theorems in Analysis by Pólya and Szegő (this formula appears, unnatributed, in many Calculus texts). While he was still at school he was influenced by the work of Srinivasa Aaiyangar Ramanujan and he discovered the exact analytical formula for the partition function as suggested by the works of Ramanujan, however, this result was first published by Hans Rademacher. Selberg makes some very interesting observations about himself and Ramanujan in his Reflections Around the Ramanujan Centenary. He studied at the University of Oslo and completed his dr. philos. (Ph.D.) grade in 1943.

During the second world war he worked in isolation due to the German military 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)=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. Then in 1948 Selberg gave an elementary proof of the prime number theorem. Paul Erdős, using a crucial result of Selberg, also obtained a proof around the same time, leading to a dispute between them about to whom this result should primarily be attributed. For all these accomplishments Selberg received the 1950 Fields Medal.

Selberg moved to the United States and settled at the Institute for Advanced Study in the 1950s where he remains today. During the 1950s he worked on introducing spectral theory into number theory, culminating into his development of the Selberg trace formula, his most famous result. This establishes a duality between the length spectrum of a 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.