Exponential sum
In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function
Therefore, a typical exponential sum may take the form
summed over a finite sequence of real numbers xn.
If we allow some real coefficients an, to get the form
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
is trivially estimated by the number N of terms. That is, the absolute value
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 reasonable to hope for is an estimate of the form
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
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 xn, to show a degree of randomness. The techniques involved are ingenious and subtle.
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