Perron's formula

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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 arithmetical 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 c > 0, c > σ, and x > 0 real, but otherwise arbitrary.

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 an special case of the Mellin discrete convolution

where and the Mellin transform. The perron formula is just the especial case of the test function Heaviside step function

References[edit]