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A finite sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942501.png" /> of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942502.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942503.png" /> is an integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942504.png" /> is a real-valued function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942505.png" />. More general sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942506.png" /> of the following form are also called trigonometric sums:
+
A finite sum  $  S $
 +
of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942507.png" /></td> </tr></table>
+
$$
 +
= \sum _ {x = 1 } ^ { P }  e ^ {2 \pi iF ( x) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942508.png" /> is a real-valued function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t0942509.png" /> is an arbitrary complex-valued function.
+
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:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425010.png" /> is a polynomial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425011.png" /> is called a [[Weyl sum|Weyl sum]]; if the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425012.png" /> has rational coefficients,
+
$$
 +
\widetilde{S}  = \
 +
\sum _ { x _ {1} = 1 } ^ { {P _ 1 } } \dots
 +
\sum _ { x _ {r} = 1 } ^ { {P _ r } }
 +
\Phi ( x _ {1} \dots x _ {r} )
 +
e ^ {2 \pi iF ( x _ {1} \dots x _ {r} ) } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425013.png" /></td> </tr></table>
+
where  $  F $
 +
is a real-valued function and  $  \Phi $
 +
is an arbitrary complex-valued function.
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425014.png" /> is called a rational trigonometric sum; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425016.png" /> is called a complete trigonometric sum; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425018.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425019.png" /> is a prime number while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425020.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425021.png" /> is a composite number, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425022.png" /> is called a trigonometric sum over prime numbers; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425025.png" /> is a polynomial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425026.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425028.png" />.
+
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====
 
====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>
 
<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
  
====Comments====
+
\frac{ax  ^ {2} }{q}
Instead of "trigonometric sum" one also uses "exponential sumexponential sum" . A complete rational exponential sum of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425029.png" />,
+
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425030.png" /></td> </tr></table>
+
is called a [[Gauss sum|Gauss sum]]. A Kloosterman sum is an exponential sum of the form
  
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 (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425031.png" /></td> </tr></table>
+
\frac{2 \pi i }{q}
 +
\left ( ux +
 +
\frac{v}{x}
 +
\right ) \right ) ,\ \
 +
u , v \in \mathbf Z .
 +
$$
  
For these there is Weil's estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094250/t09425032.png" />.
+
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]]].
 
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]]].

Latest revision as of 08:26, 6 June 2020


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
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