Atle Selberg

Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950 and an honorary Abel Prize in 2002. Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his three brothers, Sigmund and Henrik, were also mathematicians. His other brother, Arne, was 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. He studied at the University of Oslo and completed his PhD in 1943. 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 . During the war, he fought against the German invasion of Norway, and was imprisoned several times. 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 where for primes . 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.

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Atle Selberg

Automorphic form

In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms.

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.