Perron's formula

Short description: Formula to calculate the sum of an arithmetic function in analytic number theory

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

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

Here, the 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

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

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.

Generalizations

Perron's formula is just a special case of the Mellin discrete convolution

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

where

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

and

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

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

References

• Page 243 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3 
• Weisstein, Eric W.. "Perron's formula". http://mathworld.wolfram.com/PerronsFormula.html. 
• Tenenbaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. 46. Cambridge: Cambridge University Press. ISBN 0-521-41261-7.