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:
“The history of modern art is also the history of the progressive loss of art’s audience. Art has increasingly become the concern of the artist and the bafflement of the public.”
—Henry Geldzahler (1935–1994)
“It is true that this man was nothing but an elemental force in motion, directed and rendered more effective by extreme cunning and by a relentless tactical clairvoyance .... Hitler was history in its purest form.”
—Albert Camus (1913–1960)
“the future is simply nothing at all. Nothing has happened to the present by becoming past except that fresh slices of existence have been added to the total history of the world. The past is thus as real as the present.”
—Charlie Dunbar Broad (1887–1971)