01 · A life in mathematics
The person behind the work
Atle Selberg was born in Langesund, Norway, in 1917. He investigated prime numbers through analytic estimates, sieve methods and the zeros of the zeta function. His 1950 Fields Medal recognised achievements including the elementary approach to the prime number theorem associated with his and Paul Erdős’s work. He was at the Institute for Advanced Study in Princeton during the award period and became a professor there in 1951. His later trace formula linked spectral information with geometry, extending a career notable for finding powerful structures behind arithmetic counting problems.
The contribution, explained
Let π(x) count the primes not exceeding x. The prime number theorem says that π(x) is asymptotic to x/log x: the ratio approaches one as x grows. Earlier proofs used complex analysis of the zeta function. Selberg’s work, together with Erdős’s contributions, provided an elementary route using arithmetic identities and estimates instead. Elementary here describes the permitted methods, not the difficulty. The achievement showed that a global statistical law about primes could be reached by reorganising sums and controlling their errors within a different mathematical toolkit.
Why it matters
The elementary proof broadened the available methods of analytic number theory, while Selberg’s sieve and related estimates sharpened the study of primes and arithmetic sequences.