Jacob's ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\vp_2(t))|^4|\zeta(1/2+it)|^4$

TL;DR

This paper establishes a new asymptotic formula revealing a fine correlation between the fourth powers of the Riemann zeta function evaluated at specific points related to Jacob's ladders, which cannot be derived from existing theories.

Contribution

It introduces a novel asymptotic formula demonstrating a correlation between zeta function values at Jacob's ladder points, extending beyond known methods.

Findings

Established a correlation between $|{ ext{zeta}}{(1/2+i ext{vp}_2(t))}|^4$ and $| ext{zeta}(1/2+it)|^4$

Derived a new asymptotic formula not obtainable by existing theories

Enhanced understanding of the behavior of zeta function at Jacob's ladder points

Abstract

It is proved in this paper that there is a fine correlation between the values of $|\zeta(1/2+i\vp_2(t))|^4$ and $|\zeta(1/2+it)|^4$ where $\vp_2(t)$ stands for the Jacob's ladder of the second order. This new asymptotic formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.