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:
“Every member of the family of the future will be a producer of some kind and in some degree. The only one who will have the right of exemption will be the mother ...”
—Ruth C. D. Havens, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 13, by Susan B. Anthony and Ida Husted Harper (1902)
“There is a history in all men’s lives,
Figuring the natures of the times deceased,
The which observed, a man may prophesy,
With a near aim, of the main chance of things
As yet not come to life.”
—William Shakespeare (1564–1616)
“Anything in history or nature that can be described as changing steadily can be seen as heading toward catastrophe.”
—Susan Sontag (b. 1933)