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
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“... that there is no other way,
That the history of creation proceeds according to
Stringent laws, and that things
Do get done in this way, but never the things
We set out to accomplish and wanted so desperately
To see come into being.”
—John Ashbery (b. 1927)
“It may be well to remember that the highest level of moral aspiration recorded in history was reached by a few ancient Jews—Micah, Isaiah, and the rest—who took no count whatever of what might not happen to them after death. It is not obvious to me why the same point should not by and by be reached by the Gentiles.”
—Thomas Henry Huxley (1825–95)
“History ... is, indeed, little more than the register of the crimes, follies, and misfortunes of mankind.
But what experience and history teach is this—that peoples and governments have never learned anything from history, or acted on principles deduced from it.”
—Georg Wilhelm Friedrich Hegel (1770–1831)