 Perron's formula

Perron's formula recovers coefficient partial sums from a Dirichlet series by the Mellin inversion formula. Suppose $F(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ is absolutely convergent at a fixed real $c>0$. For $x>0$,The prime on the sum means that the term with $n=x$ receives weight $1/2$ if $x$ is an integer; for noninteger $x$ it is the ordinary sum over $n<x$. The integral is taken as a symmetric vertical limit, not asserted absolutely convergent on the infinite line.

The scalar truncated Perron kernel estimate tends to $1$ when $x/n>1$, to $0$ when $x/n<1$, and to $1/2$ when $x/n=1$. Absolute convergence on the line allows the coefficient sum to be interchanged with the finite integral and justifies passing to this limit. In analytic number theory, one then shifts the contour through poles and past controlled singularities: their residues supply main terms and error terms in the summatory function. The truncated Perron formula makes the finite-height error explicit.

Perron's formula is a result in analytic number theory that provides a way to express the sum of the count of integer solutions to certain equations involving prime numbers. It specifically relates to the distribution of prime numbers and is often applied in studies of prime power distributions. The formula is closely associated with the theory of Dirichlet series and often comes up in the context of additive number theory.