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.


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 uniformly convergent for \Re(s)>\sigma. Then Perron's formula is

 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.\;

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 c > 0, c > σ, and x > 0 real, but otherwise arbitrary.


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

 g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s} }=s\int_{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 Perron's formula.


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


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.