Perron's formula

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

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

$A(x)={\sum _{n\leq x}}'a(n)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }g(z){\frac {x^{z}}{z}}\,dz.$

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 that c > 0, c > σ, and x > 0.

Proof

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

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

This is nothing but a Laplace transform under the variable change $x=e^{t}.$ Inverting it one gets Perron's formula.

Proofs of Perron's formula have been published by Tom M. Apostol and by Gérald Tenenbaum.

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

$A(x)=\sum _{n\leq x}\chi (n)$

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

Generalizations

Perron's formula is a special case of the formula

$\sum _{n=1}^{\infty }a(n)f(n/x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }F(s)G(s)x^{s}ds$

where

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

and

$F(s)=\int _{0}^{\infty }f(x)x^{s-1}dx$

the Mellin transform. The Perron formula is the special case of the test function $f(1/x)=\theta (x-1),$ for $\theta (x)$ the Heaviside step function.

External links

• Weisstein, Eric W. "Perron's formula". MathWorld.
• Moreta, Jose Javier Garcia (2024), Discrete Mellin Convolution and its Extensions, Perron Formula and Explicit Formulae, General Science Journal, ISSN:1916-5382