Trigonometric sum

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A finite sum of the form

where is an integer and is a real-valued function of . More general sums of the following form are also called trigonometric sums:

where is a real-valued function and is an arbitrary complex-valued function.

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

then is called a rational trigonometric sum; if , then is called a complete trigonometric sum; if and when is a prime number while when is a composite number, then is called a trigonometric sum over prime numbers; if , and is a polynomial, then 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 and .


[1] I.M. Vinogradov, , Selected works , Springer (1985) (Translated from Russian)
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[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)
[5] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)
[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 ,

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

For these there is Weil's estimate .

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].


[a1] N.M. Korobov, "Exponential sums and their applications" , Kluwer (1992) (Translated from Russian)
[a2] N.M. Katz, "Sommes exponentielles" , Soc. Math. France (1980)
[a3] N.M. Katz, "Gauss sums, Kloosterman sums, and monodromy groups" , Princeton Univ. Press (1988)
How to Cite This Entry:
Trigonometric sum. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article