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In this paper we give new consequences that follow from our formula forincrements of the Hardy-Littlewood integral. Main of these consequences is an$\zeta$-equivalent of the Fermat-Wiles theorem. It is expressed purely by meansof the Riemann's zeta-function on the critical line and by the Jacob's ladder.

Jacob's ladders, almost linear increments of the Hardy-Littlewood  integral (1918) and their relations to the Selberg's formula (1946) and the  Fermat-Wiles theorem