On a mean-value theorem for multiple trigonometric sums

A mean-value theorem for multiple trigonometric generalizing from the G. I. Arkhipov's theorem [12, 13] was proved. The first theorem of the similar type lies in the core of the I. M. Vinogradov's method [2]. In the paper the version of theorem with “similar” lengths of changing intervals of variables. Estimates of zeta-sums of the form
$$ \sum_{n\leq P}n^{it}. $$
are the interesting application of the I.M.Vinogradov's method. The similar application of the mean-value theorem proving by us serve the estimate of sums of the form
$$ \sum_{n\leq P_1}\dots\sum_{n\leq P_r}(n_1\dots n_r+k)^{it}, \sum_{n\leq P}\tau_s(n)(n+k)^{it}, \sum_{p\leq P}(p+k)^{it}. $$