Dirichlet series
Coefficient inversion (integral formula)

For all positive integers x ≥ 1 {\displaystyle x\geq 1}, the function f at x, f (x) {\displaystyle f(x)}, can be recovered from the Dirichlet generating function (DGF) F of f (or the Dirichlet series over f) using the following integral formula whenever σ > σ a, f {\displaystyle \sigma >\sigma _{a,f}}, the abscissa of absolute convergence of the DGF F

It is also possible to invert the Mellin transform of the summatory function of f that defines the DGF F of f to obtain the coefficients of the Dirichlet series (see section below). In this case, we arrive at a complex contour integral formula related to Perron's theorem. Practically speaking, the rates of convergence of the above formula as a function of T are variable, and if the Dirichlet series F is sensitive to sign changes as a slowly converging series, it may require very large T to approximate the coefficients of F using this formula without taking the formal limit.

Another variant of the previous formula stated in Apostol's book provides an integral formula for an alternate sum in the following form for c, x > 0 {\displaystyle c,x>0} and any real ℜ (s) ≡ σ > σ a, f − c {\displaystyle \Re (s)\equiv \sigma >\sigma _{a,f}-c} where we denote ℜ (s):= σ {\displaystyle \Re (s):=\sigma }:
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