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.

[2][3] This challenged the widely held view of his time that certain theorems are only obtainable with the advanced methods of complex analysis.

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.

[4] Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between the two mathematicians.

Selberg moved to the United States and worked as an associate professor at Syracuse University and later settled at the Institute for Advanced Study in Princeton, New Jersey in the 1950s, where he remained until his death.