  In the present paper we obtain a necessary and sufficient condition to prove
the Riemann hypothesis in terms of certain properties of local extrema of the
function $\Xi(t)=\xi(\tfrac{1}{2}+it)$. First, we prove that positivity of all
local maxima and negativity of all local minima of $\Xi(t)$ form a necessary
condition for the Riemann hypothesis to be true. After showing that any
extremum point of $\Xi(t)$ is a saddle point of the function $\Re\{\xi(s)\}$,
we prove that the above properties of local extrema of $\Xi(t)$ are also a
sufficient condition for the Riemann hypothesis to hold at $t\gg 1$.
  We present a numerical example to illustrate our approach towards a possible
proof of the Riemann hypothesis. Thus, the task of proving the Riemann
hypothesis is reduced to the one of showing the above properties of local
extrema of $\Xi(t)$.
