Mean-Value Theorem for Multiple Trigonometric Sums on the Sequence of Bell Polynomials

A mean-value theorem for multiple trigonometric (exponential) sums on the sequence of Bell polynomials is proved. It generalizes I. M. Vinogradov's and G. I. Arkhipov's theorems. As is well known, a mean-value theorem of this type is at the core of Vinogradov's method. The Bell polynomials are very closely related to the Faà di Bruno theorem on higher order derivatives of a composite function. As an application of the mean-value theorem proved in the paper, estimates for the sums $\sum _{n_1\leq P}\dots \sum _{n_r\leq P}e^{2\pi i(\alpha _1Y_1(n_1)+\dots +\alpha _rY_r(n_1,\dots ,n_r))}$ are obtained, where $\alpha _s$ are real numbers and $Y_s(n_1,\dots ,n_s)$ are the degree $s$ Bell polynomials, $1\leq s\leq r$.