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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Vinogradov, , ''Selected works'' , Springer (1985) (Translated from Russian) {{MR|0807530}} {{ZBL|0577.01049}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) {{MR|0603100}} {{MR|0409380}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.M. Vinogradov, "Basic variants of the method of trigonometric sums" , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|0083.03601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) {{MR|0046485}} {{ZBL|0042.07901}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G.I. Archipov, A.A. Karatsuba, V.N. Chubarikov, "Multiple trigonometric sums" , Amer. Math. Soc. (1982) (Translated from Russian)</TD></TR></table>

====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|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|Trigonometric sums, method of]]), exponential sums play important roles in other fields, e.g. algebraic geometry, modular functions, quadrature formulas, monodromy, [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.M. Korobov, "Exponential sums and their applications" , Kluwer (1992) (Translated from Russian) {{MR|1162539}} {{ZBL|0754.11022}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.M. Katz, "Sommes exponentielles" , Soc. Math. France (1980) {{MR|0617009}} {{ZBL|0469.12007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> N.M. Katz, "Gauss sums, Kloosterman sums, and monodromy groups" , Princeton Univ. Press (1988) {{MR|0955052}} {{ZBL|0675.14004}} </TD></TR></table>