Jacob's ladders and the oscillations of the function $|\zeta(1/2+it)|^2$ around its mean-value; law of the almost exact equality of corresponding areas

TL;DR

This paper investigates the oscillations of the squared modulus of the Riemann zeta function around its mean-value, demonstrating an almost exact equality of areas that cannot be achieved by previous methods.

Contribution

It introduces a novel approach to analyze the oscillations of $| zeta(1/2+it)|^2$ and establishes an almost exact area equality, advancing understanding beyond existing techniques.

Findings

Proves an almost exact equality of areas under oscillations

Demonstrates limitations of previous methods

Provides new insights into the behavior of the zeta function

Abstract

The oscillations of the function $Z^2(t),\ t\in [0,T]$ around the main part $\sigma(T)$ of its mean-value are studied in this paper. It is proved that an almost equality of the corresponding areas holds true. This result cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.