  One classical topic in the study of local cohomology is whether the
non-vanishing of a specific local cohomology module is equivalent to the
vanishing of its annihilator; this has been studied by several authors,
including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by
Lynch and Zhang, the goal of this paper is to provide some new results about
this topic, which provide some partial positive answers to these questions. The
main technical tool we exploit is the structure of local cohomology as module
over rings of differential operators.
