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It is proved in this paper that the Jacob's ladders together with the A.Selberg's classical formula (1942) lead to a new kind of formulae for someshort trigonometric sums. These formulae cannot be obtained in the classicaltheory of A. Selberg, and all the less, in the theories of Balasubramanian,Heath-Brown and Ivic.

Jacob's ladders and some new consequences from A. Selberg's formula