Vinogradov integral

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).