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,.

Read more about this topic:  ATS Theorem

Famous quotes containing the word remark:

    “The Cretans are always liars, evil beasts, slow bellies.”
    —Bible: New Testament St. Paul, in Titus, 1:12.

    Cited by Paul, this remark is attributed to Epimenides.

    “I know not whether the remark is to our honour or otherwise, that lessons of wisdom have never such power over us, as when they are wrought into the heart, through the ground-work of a story which engages the passions: Is it that we are like iron, and must first be heated before we can be wrought upon?”
    —Laurence Sterne (1713–1768)

    “Alice felt dreadfully puzzled. The Hatter’s remark seemed to her to have no sort of meaning in it, and yet it was certainly English. “I don’t quite understand you,” she said, as politely as she could.”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)