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

TL;DR

This paper links the Riemann hypothesis to the behavior of local extrema of the $\xi(t)$ function, proposing that their positivity and negativity conditions are both necessary and sufficient for the hypothesis to hold.

Contribution

It establishes a new criterion based on local extrema properties of $\xi(t)$ that could potentially prove the Riemann hypothesis, connecting extremal behavior to the hypothesis's validity.

Findings

Positivity of all local maxima and negativity of all local minima of $\xi(t)$ are necessary for the Riemann hypothesis.

These extremal properties are also sufficient for the hypothesis at large $t$, under certain conditions.

Numerical examples support the approach towards a potential proof.

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 $\Xi(t)$.