History
If the sum is of the form
where ƒ is a smooth function, you could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum. Major advances in the subject were Van der Corput's method (c. 1920), related to the principle of stationary phase, and the later Vinogradov method (c.1930).
The large sieve method (c.1960), the work of many researchers, is a relatively transparent general principle; but no one method has general application.
Read more about this topic: Exponential Sum
Famous quotes containing the word history:
“In the history of the United States, there is no continuity at all. You can cut through it anywhere and nothing on this side of the cut has anything to do with anything on the other side.”
—Henry Brooks Adams (1838–1918)
“Yet poetry, though the last and finest result, is a natural fruit. As naturally as the oak bears an acorn, and the vine a gourd, man bears a poem, either spoken or done. It is the chief and most memorable success, for history is but a prose narrative of poetic deeds.”
—Henry David Thoreau (1817–1862)
“Jesus Christ belonged to the true race of the prophets. He saw with an open eye the mystery of the soul. Drawn by its severe harmony, ravished with its beauty, he lived in it, and had his being there. Alone in all history he estimated the greatness of man.”
—Ralph Waldo Emerson (1803–1882)