A minimal integral of the Riemann $\Xi$-function

TL;DR

This paper introduces an equilibrium sequence for the Riemann xi-function, balancing the areas of its positive and negative parts over specific intervals, contributing to understanding its oscillatory behavior.

Contribution

It presents a novel construction of an equilibrium sequence for the Riemann xi-function, balancing areas of positive and negative parts over intervals.

Findings

Existence of an equilibrium sequence omega_n for the xi-function.

Equal area property between positive and negative parts on intervals [omega_n,omega_{n+1}].

New insights into the oscillatory nature of the xi-function.

Abstract

In this paper we obtain an equilibrium sequence $\{\omega_n\}$ for which the following holds true: the areas (measures) of the figures corresponding to the positive and negative parts, respectively, of the graph of the function $\Xi(t),\ t\in[\omega_n,\omega_{n+1}]$ are equal. Dedicated to the 500th anniversary of rabbi L\"ow.