Geodesic
Jacob's Ladders and the Almost Exact Asymptotic Representation of the Hardy--Littlewood Integral
Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 446-455

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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.
Keywords: Hardy–Littlewood integral, Riemann zeta function, Gauss logarithmic integral, nonlinear integral equation, Jacob's ladder, Bonnet's mean-value theorem. 
@article{MZM_2010_88_3_a12,
     author = {J. Moser},
     title = {Jacob's {Ladders} and the {Almost} {Exact} {Asymptotic} {Representation} of the {Hardy--Littlewood} {Integral}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {446--455},
     publisher = {mathdoc},
     volume = {88},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a12/}
}
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J. Moser. Jacob's Ladders and the Almost Exact Asymptotic Representation of the Hardy--Littlewood Integral. Matematičeskie zametki, Tome 88 (2010) no. 3, pp. 446-455. http://geodesic.mathdoc.fr/item/MZM_2010_88_3_a12/