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

V. N. Chubarikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 301-310

Résumé

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, $1\leq s\leq r$.

Détail

Keywords: mean-value theorems of Vinogradov and Arkhipov, sequence of Bell polynomials, Faà di Bruno theorem.