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

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.