Exponential sum

e(x) = exp(2πix).

Therefore a typical exponential sum may take the form

Σ e(x_n)

summed over a finite sequence of real numbers

x_n.

If we allow some real coefficients a_n, to get the form

Σ a_ne(x_n),

it is the same as allowing exponents that are complex numbers. Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation.

The main thrust of the subject is that a sum

S = Σ e(x_n)

is trivially estimated by the number N of terms. That is, the absolute value

|S| ≤ N

by the triangle inequality, since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the unit circle is not of numbers all with the same argument. The best that is it reasonable to hope for, is an estimate

|S| = O(√N)

which signifies, up to the implied constant in the big O notation, that the sum resembles a random walk in two dimensions.

Such an estimate can be considered ideal; it is unattainable in many of the major problems, and estimates

|S| = o(N)

have to be used, where the o(N) function represents only a small saving on the trivial estimate. A typical 'small saving' may be a factor of log N, for example. Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence x_n, to show a degree of randomness. The techniques involved are ingenious and subtle.

Major advances in the subject were Van de 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.