# Trigonometric sum

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A finite sum $S$ of the form

$$S = \sum _ {x = 1 } ^ { P } e ^ {2 \pi iF ( x) } ,$$

where $P \geq 1$ is an integer and $F$ is a real-valued function of $x$. More general sums $\widetilde{S}$ of the following form are also called trigonometric sums:

$$\widetilde{S} = \ \sum _ { x _ {1} = 1 } ^ { {P _ 1 } } \dots \sum _ { x _ {r} = 1 } ^ { {P _ r } } \Phi ( x _ {1} \dots x _ {r} ) e ^ {2 \pi iF ( x _ {1} \dots x _ {r} ) } ,$$

where $F$ is a real-valued function and $\Phi$ is an arbitrary complex-valued function.

If $F$ is a polynomial, then $S$ is called a Weyl sum; if the polynomial $F$ has rational coefficients,

$$F ( x) = \ \frac{a _ {n} x ^ {n} + \dots + a _ {1} x }{q } ,\ \ ( a _ {n} \dots a _ {1} , q) = 1,$$

then $S$ is called a rational trigonometric sum; if $P = q$, then $\widetilde{S}$ is called a complete trigonometric sum; if $r = 1$ and $\Phi ( x _ {1} ) = 1$ when $x _ {1}$ is a prime number while $\Phi ( x _ {1} ) = 0$ when $x _ {1}$ is a composite number, then $S$ is called a trigonometric sum over prime numbers; if $r \geq 1$, $\Phi \equiv 1$ and $F$ is a polynomial, then $\widetilde{S}$ is called a multiple Weyl sum. A basic problem in the theory of trigonometric sums is that of finding upper bounds for the moduli of $S$ and $\widetilde{S}$.

#### References

 [1] I.M. Vinogradov, , Selected works , Springer (1985) (Translated from Russian) MR0807530 Zbl 0577.01049 [2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) MR0603100 MR0409380 [3] I.M. Vinogradov, "Basic variants of the method of trigonometric sums" , Moscow (1976) (In Russian) [4] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 , Teubner (1959) (Heft 13, Teil 1) Zbl 0083.03601 [5] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) MR0046485 Zbl 0042.07901 [6] G.I. Archipov, A.A. Karatsuba, V.N. Chubarikov, "Multiple trigonometric sums" , Amer. Math. Soc. (1982) (Translated from Russian)

Instead of "trigonometric sum" one also uses "exponential sumexponential sum" . A complete rational exponential sum of degree $2$,

$$S( q) = \sum _ { x= } 1 ^ { q } e ^ {2 \pi i \frac{ax ^ {2} }{q} } ,$$

is called a Gauss sum. A Kloosterman sum is an exponential sum of the form

$$K( u, v, q ) = \sum _ {\begin{array}{c} x= 1 \\ ( x, q)= 1 \end{array} } \mathop{\rm exp} \left ( \frac{2 \pi i }{q} \left ( ux + \frac{v}{x} \right ) \right ) ,\ \ u , v \in \mathbf Z .$$

For these there is Weil's estimate $| K( u, v, p ) | \leq 2 \sqrt p$.

Besides in number theory (cf. also Trigonometric sums, method of), exponential sums play important roles in other fields, e.g. algebraic geometry, modular functions, quadrature formulas, monodromy, [a1], [a2], [a3].

#### References

 [a1] N.M. Korobov, "Exponential sums and their applications" , Kluwer (1992) (Translated from Russian) MR1162539 Zbl 0754.11022 [a2] N.M. Katz, "Sommes exponentielles" , Soc. Math. France (1980) MR0617009 Zbl 0469.12007 [a3] N.M. Katz, "Gauss sums, Kloosterman sums, and monodromy groups" , Princeton Univ. Press (1988) MR0955052 Zbl 0675.14004
How to Cite This Entry:
Trigonometric sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonometric_sum&oldid=49038
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article