Recovery of a function from the coefficients of its Dirichlet series

Let L(λ) be an entire function of exponential type, letγ(t) be the function associated with L(λ) in the sense of Borel, let \(\bar D\) be the smallest closed convex set containing all the singular points ofγ(t), let λ₀, λ₁, ..., λ_n, ... be the simple zeros of L(λ), and let A \(\bar D\) be the space of functions analytic on \(\bar D\) with the topology of the inductive limit. With an arbitraryf (z) ∈ A( \(\bar D\) ) we can associate the series whereC is a closed contour containing \(\bar D\) , on and inside of whichf (z) is analytic. We give a method of recoveringf (z) from the Dirichlet coefficientsa _n.