Perron's formula

Short description: Formula to calculate the sum of an arithmetic function in analytic number theory

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 arithmetic function, by means of an inverse Mellin transform.

Statement

Let [math]\displaystyle{ \{a(n)\} }[/math] be an arithmetic function, and let

[math]\displaystyle{ g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} }[/math]

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for [math]\displaystyle{ \Re(s)\gt \sigma }[/math]. Then Perron's formula is

[math]\displaystyle{ A(x) = {\sum_{n\le x}}' a(n) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(z)\frac{x^{z}}{z} \,dz. }[/math]

Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an integer. The integral is not a convergent Lebesgue integral; it is understood as the Cauchy principal value. The formula requires that c > 0, c > σ, and x > 0.

Proof

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

[math]\displaystyle{ g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{1}^{\infty} A(x)x^{-(s+1) } dx. }[/math]

This is nothing but a Laplace transform under the variable change [math]\displaystyle{ x = e^t. }[/math] Inverting it one gets 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:

[math]\displaystyle{ \zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx }[/math]

and a similar formula for Dirichlet L-functions:

[math]\displaystyle{ L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx }[/math]

where

[math]\displaystyle{ A(x)=\sum_{n\le x} \chi(n) }[/math]

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

Generalizations

Perron's formula is just a special case of the Mellin discrete convolution

[math]\displaystyle{ \sum_{n=1}^{\infty} a(n)f(n/x)= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty}F(s)G(s)x^{s}ds }[/math]

where

[math]\displaystyle{ G(s)= \sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} }[/math]

and

[math]\displaystyle{ F(s)= \int_{0}^{\infty}f(x)x^{s-1}dx }[/math]

the Mellin transform. The Perron formula is just the special case of the test function [math]\displaystyle{ f(1/x)=\theta (x-1), }[/math] for [math]\displaystyle{ \theta(x) }[/math] the Heaviside step function.

References

• Page 243 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3 
• Weisstein, Eric W.. "Perron's formula". http://mathworld.wolfram.com/PerronsFormula.html. 
• Tenenbaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. 46. Cambridge: Cambridge University Press. ISBN 0-521-41261-7.