ATS Theorem - Remark

Remark

If the parameters and are integers, then it is possible to substitute the last relation by the following ones:

\sum_{a<k\le b} e^{2\pi i f(k)} = \int_a^be^{2\pi i f(x)}dx + \frac12e^{2\pi i f(b)} - \frac12e^{2\pi i f(a)} + \theta\frac{2\delta}{1-\delta},

where

On the applications of ATS to the problems of physics see,; see also,.

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