- Exponential sum
In

mathematics , an**exponential sum**may be a finiteFourier series (i.e. atrigonometric polynomial ), or other finite sum formed using theexponential function , usually expressed by means of the function:$e(x)\; =\; exp(2pi\; ix)$.

Therefore a typical exponential sum may take the form

:$sum\; e(x\_n)$,

summed over a finite sequence of

real number s "x"_{"n"}.**Formulation**If we allow some real coefficients "a"

_{"n"}, to get the form:$sum\; a\_n\; e(x\_n)$

it is the same as allowing exponents that are

complex number s. Both forms are certainly useful in applications. A large part oftwentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work ofHermann Weyl indiophantine approximation .**Estimates**The main thrust of the subject is that a sum

:$S=sum\; e(x\_n)$

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

absolute value :$|S|\; le\; N$

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 theunit circle is not of numbers all with the same argument. The best that is reasonable to hope for is an estimate of the form:$|S|=\; O(sqrt\{N\})$

which signifies, up to the implied constant in the

big O notation , that the sum resembles arandom 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 ofrandomness . The techniques involved are ingenious and subtle.A variant of 'Weyl differencing' investigated by Weyl involving a Generating exponential sum

$G(\; au)=\; sum\; e^\{iaf(x)+ia\; au\; n\}$

Was previosuly studied by Weyl himself, he developed a method to express the sum as the value $G(0)$ , where 'G' can be defined via a linear differential equation similar to

Dyson equation ] obtained via summation by parts.**History**If the sum is of the form

:$S(x)=\; e^\{ia\; f(x)\; \}$

where f 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 theprinciple 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.**Types of exponential sum**Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations.

Partial summation can be used to remove coefficients "a"_{"n"}, in many cases.A basic distinction is between a

**complete exponential sum**, which is typically a sum over allresidue class es "modulo" some integer "N" (or more generalfinite ring ), and an**incomplete exponential sum**where the range of summation is restricted by someinequality . Examples of complete exponential sums areGauss sum s andKloosterman sum s; these are in some sensefinite field or finite ring analogues of thegamma function and some sort ofBessel function , respectively, and have many 'structural' properties. An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by Gauss). Here there are good estimates for sums over shorter ranges than the whole set of residue classes, because, in geometric terms, the partial sums approximate aCornu spiral ; this implies massive cancellation.Auxiliary types of sums occur in the theory, for example

character sum s; going back toHarold Davenport 's thesis. TheWeil conjecture s had major applications to complete sums with domain restricted by polynomial conditions (i.e., along analgebraic variety over a finite field).One of the most general types of exponential sum is the

**Weyl sum**, with exponents 2π"if"("n") where "f" is a fairly general real-valuedsmooth function . These are the sums implicated in the distribution of the values:"f"("n") modulo 1,

according to

Weyl's equidistribution criterion . A basic advance wasWeyl's inequality for such sums, for polynomial "f".There is a general theory of

exponent pair s, which formulates estimates. An important case is where "f" is logarithmic, in relation with theRiemann zeta function . See alsoequidistribution theorem .**Example: the quadratic Gauss sum**Let $p$ be an odd prime and let $xi\; =\; e^\{2pi\; i\; /\; p\}$. Then the quadratic

Gauss sum is given by:$sum\_\{n=0\}^\{p-1\}xi^\{n^2\}\; =\; left\{\; egin\{array\}\{ll\}sqrt\{p\},\; p\; =\; 1\; mod\; 4\; \backslash isqrt\{p\},\; p\; =\; 3\; mod\; 4end\{array\}\; ight.$ where the square roots are taken to be positive.This is the ideal degree of cancellation one could hope for without any "a priori" knowledge of the structure of the sum, since it matches the scaling of a

random walk .**ee also***

Hua's lemma **External links*** [

*http://mathworld.wolfram.com/WeylSum.html A brief introduction to Weyl sums on Mathworld*]

*Wikimedia Foundation.
2010.*