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.

tatement

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

:$A(x)\; =\; \{sum\_\{nle\; x^\{star\}\; frac\{a(n)\}\{n^s\}\; =frac\{1\}\{2pi\; i\}int\_\{c-iinfty\}^\{c+iinfty\}\; g(s+z)frac\{x^\{z\{z\}\; dz;$

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 $c>0$ 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

:$g(s)=sum\_\{n=1\}^\{infty\}\; frac\{a(n)\}\{n^\{s\}\; \}=sint\_\{0\}^\{infty\}\; A(x)x^\{-(s+1)\; \}\; dx.$

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:

:$zeta(s)=sint\_1^infty\; frac\{lfloor\; x\; floor\}\{x^\{s+1,dx$

and a similar formula for Dirichlet L-functions:

:$L(s,chi)=sint\_1^infty\; frac\{A(x)\}\{x^\{s+1,dx$

where

:$A(x)=sum\_\{nle\; x\}\; chi(n)$

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

References

* Page 243 of Apostol IANT
*
* Tenebaum, Gérald (1995). "Introduction to analytic and probabilistic number theory", Cambridge University Press, Cambridge. ISBN 0521412617.