Jacob's ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral

Abstract

In this paper we introduce a nonlinear integral equation such that the system of global solution to this equation represents a class of a very narrow beam at $T\to\infty$ (an analogue to the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral. The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown and Ivic.