Selberg's identity

In number theory, Selberg's identity is an approximate identity involving logarithms of primes named after Atle Selberg. The identity, discovered jointly by Selberg and Paul Erdős, was used in the first elementary proof for the prime number theorem.

Statement

There are several different but equivalent forms of Selberg's identity. One form is

${\displaystyle \sum _{p<x}(\log p)^{2}+\sum _{pq<x}\log p\log q=2x\log x+O(x)}$

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

${\displaystyle \sum _{n<x}c_{n}}$

where the numbers

${\displaystyle c_{n}=\Lambda (n)\log n+\sum _{d\,|\,n}\Lambda (d)\Lambda (n/d)}$

are the coefficients of the Dirichlet series

${\displaystyle {\frac {\zeta ^{\prime \prime }(s)}{\zeta (s)}}=\left({\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\right)^{\prime }+\left({\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\right)^{2}=\sum {\frac {c_{n}}{n^{s}}}.}$

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 ${\displaystyle \sum _{n<x}c_{n}.}$

Another variation of the identity

Selberg's identity sometimes also refers to the following divisor sum identity involving the von Mangoldt function and the Möbius function when ${\displaystyle n\geq 1}$:^[1]

${\displaystyle \Lambda (n)\log(n)+\sum _{d\,|\,n}\Lambda (d)\Lambda \!\left({\frac {n}{d}}\right)=\sum _{d\,|\,n}\mu (d)\log ^{2}\left({\frac {n}{d}}\right).}$

This variant of Selberg's identity is proved using the concept of taking derivatives of arithmetic functions defined by ${\displaystyle f^{\prime }(n)=f(n)\cdot \log(n)}$ in Section 2.18 of Apostol's book (see also this link).