= Perron's formula
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= Perron formula
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<Perron's formula> recovers coefficient <partial sums> from a <Dirichlet series> by the <Mellin inversion formula>. Suppose $F(s)=\sum_{n\ge1}a_nn^{-s}$ is absolutely convergent at a fixed real $c>0$. For $x>0$,
$$
\sum_{n\le x}^{\prime}a_n=\lim_{T\to\infty}\frac1{2\pi i}\int_{c-iT}^{c+iT}F(s)\frac{x^s}{s}\,ds.
$$
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.
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