Perron's formula

Encyclopedia : P : PE : PER : Perron's formula

In mathematics, and more particularly in analytic number theory, Perron's formula is a formula of Oskar Perron to calculate the sum of an arithmetical function, by means of an inverse Mellin transform.

Contents

1 Statement
2 Proof
3 Examples
4 References

Statement

Let [\] be an arithmetic function, and let

[ g(s)=\sum_^ \frac} ]

be the corresponding Dirichlet series. Presume the Dirichlet series to be absolutely convergent for [\Re(s)>\sigma_a]. Then Perron's formula is

[ A(x) = }^ \frac =\frac\int_^ dz\; g(s+z)\frac} ]

Here, the star on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an integer. The formula requires [c>0] and [x>0] real, but otherwise arbitrary. The formula holds for [\Re(s)>\sigma_a - c]

Proof

An easy sketch of the proof comes from taking the Abel's sum formula

[ g(s)=\sum_^ \frac }=s\int_^ dx A(x)x^. ]

This is nothing but a Laplace transform under the variable change [x=e^t.] Inverting it one gets the Perron's formula.

Examples

Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums. Thus, for example, one has the famous integral representation for the Riemann zeta function:

[\zeta(s)=s\int_1^\infty \frac}\,dx]

and a similar formula for Dirichlet L-functions:

[L(s,\chi)=s\int_1^\infty \frac}\,dx]

where

[A(x)=\sum_ \chi(n)]

and [\chi(n)] is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldt function.

References

↑ Tom Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.
, [Perron's formula] at MathWorld.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.