Trigonometric sum

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)

Comments

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