Vinogradov theorem about the average

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

where is the average value of the trigonometric sum. It is formulated as follows. If, for a non-negative integer one sets

then, if and for an integer ,

The estimate of given by Vinogradov's theorem is asymptotically exact. The theorem is fundamental in the Vinogradov method for estimating Weyl sums (cf. Weyl sum). In addition, it yielded a number of almost optimal results in classical problems in number theory (cf. Waring problem; Hilbert–Kamke problem; Distribution modulo one of a polynomial).

References

[1]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[2]	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)