Dirichlet polynomials and a moment problem

Consider a linear functional L defined on the space $${\mathcal {D}}[s]$$ of Dirichlet polynomials with real coefficients and set {D}}_+[s]$$ non-negative elements in {D}}[s].$$ An analogue Riesz–Haviland theorem this context asks: What are all -positive functionals L, which moment functionals? Since {D}}[s],$$ when considered as subspace $$C([0, \infty ), {\mathbb {R}}),$$ fails to be an adapted sense Choquet, general form is not applicable situation. In attempt answer forgoing question, we arrive at notion sequence, call Hausdorff log-moment sequence. Apart from theorem, show that any sequence combination $$\{1, 0, \ldots , \}$$ $$\{f(\log (n)\}_{n \geqslant 1}$$ for completely monotone function $$f : [0, ) \rightarrow ).$$ Moreover, such f uniquely determined by question.