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.
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.
An easy sketch of the proof comes from taking Abel's sum formula This is nothing but a Laplace transform under the variable change
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
The Perron formula is just the special case of the test function