View source for Vinogradov theorem about the average

A theorem providing an upper bound of the value of a [[Vinogradov integral|Vinogradov integral]]:

<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/v096700/v0967001.png" /></td> </tr></table>

<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/v096700/v0967002.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967003.png" /> is the average value of the trigonometric sum. It is formulated as follows. If, for a non-negative integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967004.png" /> one sets

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then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967006.png" /> and for an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967007.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/v/v096/v096700/v0967008.png" /></td> </tr></table>

The estimate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096700/v0967009.png" /> given by Vinogradov's theorem is asymptotically exact. The theorem is fundamental in the [[Vinogradov method|Vinogradov method]] for estimating Weyl sums (cf. [[Weyl sum|Weyl sum]]). In addition, it yielded a number of almost optimal results in classical problems in number theory (cf. [[Waring problem|Waring problem]]; [[Hilbert–Kamke problem|Hilbert–Kamke problem]]; [[Distribution modulo one|Distribution modulo one]] of a polynomial).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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  (1959)  (Heft 13, Teil 1)</TD></TR></table>