On a mean-value theorem for multiple trigonometric sums
Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 341-356
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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 $$ \sum_{n\leq P_1}\dots\sum_{n\leq P_r}(n_1\dots n_r+k)^{it}, \sum_{n\leq P}\tau_s(n)(n+k)^{it}, \sum_{p\leq P}(p+k)^{it}. $$
Keywords:
the mean-value theorem of I. M. Vinigradov and G. I. Arkhipov, the multivariate divisor function, prime numbers, the zeta-sum.
@article{CHEB_2020_21_1_a21,
author = {V. N. Chubarikov},
title = {On a mean-value theorem for multiple trigonometric sums},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {341--356},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a21/}
}
V. N. Chubarikov. On a mean-value theorem for multiple trigonometric sums. Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 341-356. http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a21/