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:
“I feel as tall as you.”
—Ellis Meredith, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 14, by Susan B. Anthony and Ida Husted Harper (1902)
“Throughout the history of commercial life nobody has ever quite liked the commission man. His function is too vague, his presence always seems one too many, his profit looks too easy, and even when you admit that he has a necessary function, you feel that this function is, as it were, a personification of something that in an ethical society would not need to exist. If people could deal with one another honestly, they would not need agents.”
—Raymond Chandler (1888–1959)
“History, as an entirety, could only exist in the eyes of an observer outside it and outside the world. History only exists, in the final analysis, for God.”
—Albert Camus (1913–1960)