Perron's formula

From Wikipedia, the free encyclopedia

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 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[]

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[]

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[]

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[]

  • 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.
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