  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.
