Local Extrema of the $\Xi(t)$ Function and The Riemann Hypothesis

Abstract

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 $\Xi(t)$.