Perron's formula

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

1 Statement
2 Proof
3 Examples
4 References

[edit] Statement

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

$g(s)=\sum_{n=0}^{\infty} \frac{a(n)}{n^{s}}$

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

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

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

$\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx$