View source for Exponential sum

{{Short description|Finite sum formed using the exponential function}}
In [[Mathematics|mathematics]], an '''exponential sum''' may be a finite [[Fourier series]] (i.e. a [[Trigonometric polynomial|trigonometric polynomial]]), or other finite sum formed using the [[Exponential function|exponential function]], usually expressed by means of the function

:<math>e(x) = \exp(2\pi ix).\,</math>

Therefore, a typical exponential sum may take the form

:<math>\sum_n e(x_n),</math>

summed over a finite [[Sequence|sequence]] of [[Real number|real number]]s ''x''<sub>''n''</sub>.

==Formulation==

If we allow some real coefficients ''a''<sub>''n''</sub>, to get the form

:<math>\sum_n a_n e(x_n)</math>

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

==Estimates==

The main thrust of the subject is that a sum

:<math>S=\sum_n e(x_n)</math>

is ''trivially'' estimated by the number ''N'' of terms. That is, the [[Absolute value|absolute value]]

:<math>|S| \le N\,</math>

by the [[Triangle inequality|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|unit circle]] is not of numbers all with the same [[Parameter|argument]]. The best that is reasonable to hope for is an estimate of the form

:<math>|S|= O(\sqrt{N})\,</math>

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

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

:<math>|S|= o(N)\,</math>

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''<sub>''n''</sub>, to show a degree of [[Randomness|randomness]]. The techniques involved are ingenious and subtle.

A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum

<math> G(\tau)= \sum_n e^{iaf(x)+ia\tau n} </math>

was previously studied by Weyl himself, he developed a method to express the sum as the value <math> G(0)</math>, 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

:<math> S(x)= e^{ia f(x) } </math>

where ''ƒ'' is a smooth function, we 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.

==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''<sub>''n''</sub>, in many cases.

A basic distinction is between a '''complete exponential sum''', which is typically a sum over all residue classes ''[[Modular arithmetic|modulo]]'' some integer ''N'' (or more general [[Finite ring|finite ring]]), and an '''incomplete exponential sum''' where the range of summation is restricted by some [[Inequality (mathematics)|inequality]]. Examples of complete exponential sums are [[Gauss sum]]s and [[Kloosterman sum]]s; these are in some sense [[Finite field|finite field]] or finite ring analogues of the [[Gamma function|gamma function]] and some sort of [[Bessel 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 [[Biography:Carl Friedrich Gauss|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 a Cornu spiral; this implies massive cancellation.

Auxiliary types of sums occur in the theory, for example [[Character sum|character sum]]s; going back to [[Biography:Harold Davenport|Harold Davenport]]'s thesis. The [[Weil conjectures]] had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an [[Algebraic variety|algebraic variety]] over a finite field).

==Weyl sums==
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-valued smooth function. These are the sums involved in the distribution of the values

:''&fnof;''(''n'') modulo 1,

according to Weyl's equidistribution criterion. A basic advance was [[Weyl's inequality (number theory)|Weyl's inequality]] for such sums, for polynomial ''f''.

There is a general theory of exponent pairs, which formulates estimates. An important case is where ''f'' is logarithmic, in relation with the [[Riemann zeta function]].  See also [[Equidistribution theorem|equidistribution theorem]].<ref name=Mont39>Montgomery (1994) p.39</ref>

==Example: the quadratic Gauss sum==

Let ''p'' be an odd prime and let <math>\xi = e^{2\pi i / p}</math>. Then
the [[Quadratic Gauss sum]] is given by

:<math>\sum_{n=0}^{p-1}\xi^{n^2} =
\begin{cases}
\sqrt{p}, & p = 1 \mod 4 \\
i\sqrt{p}, & p = 3 \mod 4
\end{cases}
</math>

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|random walk]].

== Statistical model ==
The ''sum of exponentials'' is a useful model in [[Medicine:Pharmacokinetics|pharmacokinetics]] ([[Chemistry:Chemical kinetics|chemical kinetics]] in general) for describing the concentration of a substance over time. The exponential terms correspond to first-order reactions, which in pharmacology corresponds to the number of modelled diffusion ''compartments''.<ref>{{cite journal |last1=Hughes |first1=JH |last2=Upton |first2=RN |last3=Reuter |first3=SE |last4=Phelps |first4=MA |last5=Foster |first5=DJR |title=Optimising time samples for determining area under the curve of pharmacokinetic data using non-compartmental analysis. |journal=The Journal of Pharmacy and Pharmacology |date=November 2019 |volume=71 |issue=11 |pages=1635–1644 |doi=10.1111/jphp.13154 |pmid=31412422 |doi-access=}}</ref><ref>{{cite journal |last1=Hull |first1=CJ |title=Pharmacokinetics and pharmacodynamics. |journal=British Journal of Anaesthesia |date=July 1979 |volume=51 |issue=7 |pages=579–94 |doi=10.1093/bja/51.7.579 |pmid=550900|doi-access=free }}</ref>

==See also==
* [[Hua's lemma]]

==References==
{{reflist}}
* {{cite book | last=Montgomery | first=Hugh L. |  title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=American Mathematical Society | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}
* {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}

==Further reading==
* {{cite book | zbl=0754.11022 | last=Korobov | first=N.M. | title=Exponential sums and their applications | others=Translated from the Russian by Yu. N. Shakhov | series=Mathematics and Its Applications. Soviet Series. | volume=80 | location=Dordrecht | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-1647-9 }}

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

[[Category:Exponentials]]
[[Category:Analytic number theory]]

{{Sourceattribution|Exponential sum}}