# 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 $\widetilde{S}$.

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