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.
Let
\{a(n)\}
| infty | |
| g(s)=\sum | |
| n=1 |
| a(n) | |
| ns |
\Re(s)>\sigma
A(x)={\sumn\le
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.
An easy sketch of the proof comes from taking Abel's sum formula
| infty | |
| g(s)=\sum | |
| n=1 |
| a(n) | |
| ns |
| infty | |
| =s\int | |
| 1 |
A(x)x-(s+1)dx.
This is nothing but a Laplace transform under the variable change
x=et.
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:
| infty | |
| \zeta(s)=s\int | |
| 1 |
| \lfloorx\rfloor | |
| xs+1 |
dx
and a similar formula for Dirichlet L-functions:
| infty | |
| L(s,\chi)=s\int | |
| 1 |
| A(x) | |
| xs+1 |
dx
where
A(x)=\sumn\le\chi(n)
and
\chi(n)
Perron's formula is just a special case of the Mellin discrete convolution
| infty | |
| \sum | |
| n=1 |
a(n)f(n/x)=
| 1 | |
| 2\pii |
| c+iinfty | |
| \int | |
| c-iinfty |
F(s)G(s)xsds
where
G(s)=
| infty | |
| \sum | |
| n=1 |
| a(n) | |
| ns |
and
F(s)=
| infty | |
| \int | |
| 0 |
f(x)xs-1dx
the Mellin transform. The Perron formula is just the special case of the test function
f(1/x)=\theta(x-1),
\theta(x)