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The elementary geometric properties of Jacob's ladders of the second orderlead to a class of new asymptotic formulae for short and microscopic parts ofthe Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot beobtained by methods of Balasubramanian, Heath-Brown and Ivic.

Jacob's ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^4$