01 · The idea
In everyday language
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.
