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Difference between revisions of "Vinogradov integral"

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A multiple integral of the form
 
A multiple integral 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/v/v096/v096680/v0966801.png" /></td> </tr></table>
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$$ \int \limits_0^1 \dots \int \limits_0^1 |S|^{2k} d \alpha_1 \dots d \alpha_n, $$
  
 
where
 
where
  
<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/v/v096/v096680/v0966802.png" /></td> </tr></table>
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$$ S = \sum_{1 \leq x \leq P} e^{2 \pi i (\alpha_1 x + \dots + \alpha_n x^n)}, $$
  
which is the average value of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096680/v0966803.png" /> of the modulus of a [[Trigonometric sum|trigonometric sum]]. Vinogradov's theorem on the value of this integral — the theorem about the average — forms the basis of estimates of Weyl sums (cf. [[Vinogradov method|Vinogradov method]]; [[Vinogradov theorem about the average|Vinogradov theorem about the average]]).
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which is the average value of order $2k$ of the modulus of a [[Trigonometric sum|trigonometric sum]]. Vinogradov's theorem on the value of this integral — the theorem about the average — forms the basis of estimates of Weyl sums (cf. [[Vinogradov method|Vinogradov method]]; [[Vinogradov theorem about the average|Vinogradov theorem about the average]]).
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Latest revision as of 21:35, 14 January 2017

A multiple integral of the form

$$ \int \limits_0^1 \dots \int \limits_0^1 |S|^{2k} d \alpha_1 \dots d \alpha_n, $$

where

$$ S = \sum_{1 \leq x \leq P} e^{2 \pi i (\alpha_1 x + \dots + \alpha_n x^n)}, $$

which is the average value of order $2k$ of the modulus of a trigonometric sum. Vinogradov's theorem on the value of this integral — the theorem about the average — forms the basis of estimates of Weyl sums (cf. Vinogradov method; Vinogradov theorem about the average).

How to Cite This Entry:
Vinogradov integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vinogradov_integral&oldid=17597
This article was adapted from an original article by A.A. Karatsuba (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article