Atle Selberg

1917–2007, Langesund, Norway, Princeton, New Jersey, USA

Arithmetic and number theoryAnalysisFields Medal 1950Wolf Prize in Mathematics 1986

Norwegian-American mathematician, Fields Medalist 1950, celebrated for the elementary proof of the prime number theorem, the Selberg sieve, the positive proportion of zeta zeros on the critical line, and the Selberg trace formula.

Atle Selberg was born in Langesund, Norway, the youngest of nine children, and taught himself mathematics as a boy by reading Ramanujan's collected works, which he found in his father's library — an experience he called "a revelation." He studied at the University of Oslo, and in 1942, working almost in isolation under the German occupation, he proved that a positive proportion of the non-trivial zeros of the Riemann zeta function lie on the critical line — a result announced in wartime Norway that astonished the field when it became known.

In 1947 Selberg moved to the United States, and after a year at Syracuse University settled permanently at the Institute for Advanced Study in Princeton in 1949. That same year he and Paul Erdős independently found elementary proofs of the prime number theorem — the first proofs avoiding complex analysis in the half-century since Hadamard and de la Vallée Poussin — based on the asymptotic formula now called the Selberg identity. Priority and publication disputes marred the episode, but the achievement secured Selberg the Fields Medal at the 1950 Harvard congress, shared with Laurent Schwartz.

Selberg remained at the Institute for Advanced Study for the rest of his life. In 1956 he introduced the trace formula relating the lengths of closed geodesics on a hyperbolic surface to the eigenvalues of its Laplacian, a cornerstone of modern automorphic forms and spectral theory, together with the Selberg zeta function and his eigenvalue conjecture. He received the Wolf Prize in 1986 and an honorary Abel Prize at its inauguration in 2002. He died in Princeton in 2007.

Workplaces: University of Oslo, Institute for Advanced Study, Princeton

Norway, United States

Award-winning work

Fields Medal 1950Selberg received the Fields Medal for his profound work on the zeros of the Riemann zeta function, for the Selberg sieve, and for the elementary proof of the prime number theorem he gave in 1949 — independently of Paul Erdős — which removed the century-old dependence on complex analysis. His sieve method, a systematic way of bounding the count of primes and almost-primes in integer sequences, became a fundamental tool of analytic number theory. He later proved that a positive proportion of the zeros of the zeta function lie on the critical line, and went on to create the trace formula connecting spectral theory and automorphic forms.

Contributions, linked to the library

TheoremPrime number theoremGave the first elementary proof (1949), independent of Erdős, built on the Selberg identity.Library topicSieve methodsHis large sieve and the Selberg sieve are foundational tools of sieve theory.Library topicThe Riemann zeta function and the Riemann hypothesisProved that a positive proportion of zeta zeros lie on the critical line, the strongest unconditional result of its era.