World War II
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 .
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 March 1948, Selberg established, by elementary means, the asymptotic formula
where
for primes . By July of that year, Selberg and Paul Erdős had each obtained elementary proofs of the prime number theorem, both using Selberg's then unpublished asymptotic formula as a starting point. Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between the two mathematicians.
For his fundamental accomplishments during the 1940s, Selberg received the 1950 Fields Medal.
Read more about this topic: Atle Selberg
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