Difference between revisions of "Vinogradov integral"
From Encyclopedia of Mathematics
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A multiple integral of the form | 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 | 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 | + | 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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| + | {{TEX|done}} | ||
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
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