Selberg's identity

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.

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

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]

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

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).

• Pisot, Charles (1949), Démonstration élémentaire du théorème des nombres premiers, Séminaire Bourbaki, 1, MR 1605145
• Selberg, Atle (1949), "An elementary proof of the prime-number theorem", Ann. of Math., 2, 50 (2): 305–313, doi:10.2307/1969455, JSTOR 1969455, MR 0029410