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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.

Statement

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

$g(s)=\sum _{n=1}^{\mathrm{\infty }}\frac{a(n)}{n^s}$

be the corresponding Dirichlet series. Presume the Dirichlet series to be absolutely convergent for $\mathrm{\Re }(s)>{\sigma }_a$. Then Perron's formula is

$A(x)={\sum _{n\le x}}^{\star }a(n)=\frac{1}{2\pi i}\underset{c-i\mathrm{\infty }}{\overset{c+i\mathrm{\infty }}{\int }}g(z)\frac{x^z}{z}dz.$

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 Abel's sum formula

$g(s)=\sum _{n=1}^{\mathrm{\infty }}\frac{a(n)}{n^s}=s\underset{0}{\overset{\mathrm{\infty }}{\int }}A(x)x^{-(s+1)}dx.$

This is nothing but a Laplace transform under the variable change $x=e^t.$ 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:

$\zeta (s)=s\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\lfloor x\rfloor }{x^{s+1}}dx$

and a similar formula for Dirichlet L-functions:

$L(s,\chi )=s\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{A(x)}{x^{s+1}}dx$

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

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