Perron's formula

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.

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

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for ${\displaystyle \Re (s)>\sigma }$. 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.

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

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