number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, Selberg's identity is an approximate identity involving

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

s of

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

named after

Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...

. The identity, discovered jointly by Selberg and

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

, was used in the first

elementary proof In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain ...

for the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...

Statement

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

\sum_(\log p)^2 +\sum_\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 :

\sum_ c_n

where the numbers :

c_n = \Lambda(n)\log n +\sum_\Lambda(d)\Lambda(n/d)

are the coefficients of the

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...

\frac=\left(\frac\right)^\prime +\left(\frac\right)^2=\sum \frac.

This

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

has a pole of order 2 at ''s'' = 1 with coefficient 2, which gives the dominant term 2''x'' log(''x'') in the

asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...

\sum_ c_n.

Another variation of the identity

Selberg's identity sometimes also refers to the following divisor sum identity involving the

von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mang ...

and the

Möbius function The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...

when

n \geq 1

: :

\Lambda(n) \log(n) + \sum_ \Lambda(d) \Lambda\!\left(\frac\right) = \sum_ \mu(d) \log^2\left(\frac\right).

This variant of Selberg's identity is proved using the concept of taking derivatives of

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...

s defined by

f^(n) = f(n) \cdot \log(n)

in Section 2.18 of Apostol's book (see als
this link
.

References

* *{{citation, mr=0029410 , last=Selberg, first= Atle , title=An elementary proof of the prime-number theorem , journal=Ann. of Math. , series= 2, volume= 50, year=1949, issue=2, pages= 305–313, doi=10.2307/1969455, jstor=1969455 Prime numbers Mathematical identities