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

$$J_b = J_{b, n} (P) = \int \limits_0^1 \dots \int \limits_0^1 \left| \sum_{x \, = \, 1}^P e^{2 \pi i (\alpha_n x^n + \dots + \alpha_1 x)} \right|^{2b} d \alpha_n \dots d \alpha_1,$$

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

$$D_t = (20n)^{n(n + 1)t/2}, \qquad b_t = nt + \left[{\frac{n(n + 1)}{4} + 1}\right],$$

then, if $l > 0$ and for an integer $b \geq b_l$,

$$J_b = J_{b, n} (P) < D_l P^{2b - (1 + (1 - 1/n)^l)n(n + 1)/2}.$$

The estimate of $J_b$ 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)
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