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

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 $|ζ(\frac{1}{2}+i\vp_1(t))|^4|\zf|^2$. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivic. Dedicated to the 75th aniversary of Anatolii Alekseevich Karatsuba.