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.

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.

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

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

Perron's formula is just a special case of the Mellin discrete convolution

where

and