  Let $(M^{n}, g)$ be a closed connected Einstein space, $n=dim M ,$ and
$\kappa_{0} $ be the lower bound of the sectional curvature. In this paper, we
prove Udo Simon's conjecture: on closed Einstein spaces, $n\geq 3,$ there is no
eigenvalue $\lambda$ such that $$n\kappa_{0} < \lambda < 2(n + 1)\kappa_{0},$$
and both bounds are the best possible. Furthermore, we develop Simon's
conjecture to the next gap of eigenvalue $\lambda:$ on closed Einstein spaces,
there is no $\lambda$ such that $$
  2(n + 1)\kappa_{0}< \lambda < 2(n+2)\kappa_{0}, $$ and both bounds are the
best possible.
