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.

References[edit]