Perron's formula

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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.

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[edit] 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} \frac{a(n)}{n^s} =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} dz\; g(s+z)\frac{x^{z}}{z}

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

[edit] 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} dx A(x)x^{-(s+1) }.

This is nothing but a Laplace transform under the variable change x = et. Inverting it one gets the Perron's formula.

[edit] 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+1}}\,dx

and a similar formula for Dirichlet L-functions:

L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx

where

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

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

[edit] References

[edit] Multivariable generalization

A generalization of this formula to the multivariable setting as been announced by Peter Swinnerton-Dyer in 2007.

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