Perron's formula

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

Contents

Statement

Let \{a(n)\} be an arithmetic function, and let

g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}}

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) = {\sum_{n\le x}}^{\star} a(n) =\frac{1}{2\pi i}\int_{c-i\infty}^{c%2Bi\infty} g(z)\frac{x^{z}}{z} dz.\;

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>\sigma_a and x>0 real, but otherwise arbitrary.

Proof

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

g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{0}^{\infty} A(x)x^{-(s%2B1) } dx.

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{\lfloor x\rfloor}{x^{s%2B1}}\,dx

and a similar formula for Dirichlet L-functions:

L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s%2B1}}\,dx

where

A(x)=\sum_{n\le x} \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