An Application of Hausdorff Moment Problem

In this work we apply Hausdorff moment problem to prove a necessary and sufficient condition for a complex sequence to be positive. Then we apply it to a subclass of genus $0$ entire functions $f(z)$ to obtain an infinite family of necessary and sufficient conditions for $f(z)$ to have only negative roots, which in turn to give infinite many necessary and sufficient conditions for the Riemann hypothesis and the generalized Riemann hypothesis with a primitive Dirichlet character.