  Let $R$ be a commutative Noetherian ring, $\mathfrak a$ a proper ideal of $R$
and $N$ a non-zero finitely generated $R$-module with $N\neq \mathfrak a N$.
Let $d$ (respectively $c$) be the smallest (respectively greatest) non-negative
integer $i$ such that the local cohomology $\operatorname{H}^i_{\mathfrak
a}(N)$ is non-zero. In this paper, we provide sharp bounds under inclusion for
the annihilators of the local cohomology modules $\operatorname{H}^d_{\mathfrak
a}(N)$, $\operatorname{H}^c_{\mathfrak a}(N)$ and these annihilators are
computed in certain cases. Also, we construct a counterexample to Lynch's
conjecture.
