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.
There are several different but equivalent forms of Selberg's identity. One form is
\sump<x(logp)2+\sumpq<xlogplogq=2xlogx+O(x)
The strange-looking expression on the left side of Selberg's identity is (up to smaller terms) the sum
\sumn<xcn
cn=Λ(n)logn+\sumd|nΛ(d)Λ(n/d)
\zeta\prime(s) | =\left( | |
\zeta(s) |
\zeta\prime(s) | |
\zeta(s) |
\right)\prime+\left(
\zeta\prime(s) | |
\zeta(s) |
\right)2=\sum
cn | |
ns |
.
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
\sumn<xcn.
Selberg's identity sometimes also refers to the following divisor sum identity involving the von Mangoldt function and the Möbius function when
n\geq1
Λ(n)log(n)+\sumd|nΛ(d)Λ\left(
n | |
d |
\right)=\sumd|n\mu(d)
| ||||
log |
\right).
This variant of Selberg's identity is proved using the concept of taking derivatives of arithmetic functions defined by
f\prime(n)=f(n) ⋅ log(n)