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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.

Contents

Statement

Let { a ( n ) } be an arithmetic function, and let

g ( s ) = ∑ n = 1 ∞ a ( n ) n s

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for ℜ ( s ) > σ . Then Perron's formula is

A ( x ) = ∑ n ≤ x ′ a ( n ) = 1 2 π i ∫ c − i ∞ c + i ∞ g ( z ) x z z d z .

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

An easy sketch of the proof comes from taking Abel's sum formula

g ( s ) = ∑ n = 1 ∞ a ( n ) n s = s ∫ 0 ∞ A ( x ) x − ( s + 1 ) d x .

This is nothing but a Laplace transform under the variable change x = e t . 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:

ζ ( s ) = s ∫ 1 ∞ ⌊ x ⌋ x s + 1 d x

and a similar formula for Dirichlet L-functions:

L ( s , χ ) = s ∫ 1 ∞ A ( x ) x s + 1 d x

where

A ( x ) = ∑ n ≤ x χ ( n )

and χ ( n ) is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldt function.

References

Perron's formula Wikipedia
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