Jacob's ladders, logarithmic modification of the Hardy-Littlewood integral (1918), Titchmarsh's $\Omega$-theorem (1928) and new point of contact with the Fermat-Wiles theorem

TL;DR

This paper introduces new connections between Jacob's ladders and Fermat-Wiles theorem via a logarithmic modification of the Hardy-Littlewood integral, revealing asymptotic conservation laws for related areas.

Contribution

It presents novel links between Jacob's ladders and Fermat-Wiles theorem through a modified Hardy-Littlewood integral and explores associated asymptotic area laws.

Findings

Two new points of contact between Jacob's ladders and Fermat-Wiles theorem.

Asymptotic laws of conservation for areas related to the modified Hardy-Littlewood integral.

Abstract

In this paper we obtain two new points of contact between Jacob's ladders and Fermat-Wiles theorem. They are generated by a logarithmic modification of the Hardy-Littlewood integral. Furthermore, we present a kind of asymptotic laws of conservation for a set of areas connected with above mentioned modification of the Hardy-Littlewood integral.