Jacob's ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations

TL;DR

This paper introduces new formulae for the Hardy-Littlewood integral and a novel asymptotic formula for a sixth order expression involving the Riemann zeta function, advancing understanding of its intricate structure.

Contribution

It provides the first asymptotic formula for a complex sixth order expression and new formulae for the Hardy-Littlewood integral, not obtainable by existing theories.

Findings

New formulae for short and microscopic parts of the Hardy-Littlewood integral.

First asymptotic formula for the sixth order expression involving the Riemann zeta function.

Advances in the analysis of the structure of the Riemann zeta function.

Abstract

In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth order expression $|\zeta(\frac{1}{2}+i\vp_1(t))|^4|\zf|^2$. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivic. Dedicated to the 75th aniversary of Anatolii Alekseevich Karatsuba.