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
[edit]Let be an arithmetic function, and let
be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for . Then Perron's formula is
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
[edit]An easy sketch of the proof comes from taking Abel's sum formula
This is nothing but a Laplace transform under the variable change Inverting it one gets Perron's formula.
Examples
[edit]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:
and a similar formula for Dirichlet L-functions:
where
and is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldt function.
Generalizations
[edit]Perron's formula is just a special case of the Mellin discrete convolution
where
and
the Mellin transform. The Perron formula is just the special case of the test function for the Heaviside step function.
References
[edit]- 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, MR 0434929, Zbl 0335.10001
- Weisstein, Eric W. "Perron's formula". MathWorld.
- Tenenbaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. Vol. 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.