Jacob's Ladders and the Almost Exact Asymptotic Representation of the Hardy–Littlewood Integral

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as $T\to\infty$ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy–Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath–Brown, and Ivic.