In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by Selberg (1949). Selberg and Erdős both used this identity to given elementary proofs of the prime number theorem.
Contents
Statement
There are several different but equivalent forms of Selberg's identity. One form is
where the sums are over primes p and q.
Explanation
The strange-looking expression on the left side of Selberg's identity is (up to smaller terms) the sum
where the numbers
are the coefficients of the Dirichlet series
This function has a pole of order 2 at s=1 with coefficient 2, which gives the dominant term 2x log(x) in the asymptotic expansion of