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Selberg's identity

From Wikipedia, the free encyclopedia

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

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

where the sums are over primes p and q.

Explanation

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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

Another variation of the identity

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Selberg's identity sometimes also refers to the following divisor sum identity involving the von Mangoldt function and the Möbius function when :[1]

This variant of Selberg's identity is proved using the concept of taking derivatives of arithmetic functions defined by in Section 2.18 of Apostol's book (see also this link).

References

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  1. ^ Apostol, T. (1976). Introduction to Analytic Number Theory. New York: Springer. p. 46 (Section 2.19). ISBN 0-387-90163-9.