On Vinogradov’s series over primes

Abstract

In the paper theorems on the convergence trigonometrical series over primes are proven.
We continue the G. I. Arkhipov’s and K. I. Oskolkov’s investigations on spechial trigonometric series, based on the I. M. Vinogradov’s method on estimations of H. Weyl’s trigonometric sums with polynomial in the argument of the trigonometric function.
In this paper we consider special trigonometric series over prime numbers of of three types in the dependence of functions of an argument trigonometric function: the square root, the linear
and the the general polynomial from the one variable, running sequently all prime numbers.
Here we essentially are used the I. M. Vinogradov’s estimates of trigonometric sums over prime numbers.

We note the one circumstance, that appeares the need touse the holding of Grand Riemann’s hypotheses.
Besides, in the case of linear polynomials of arguments for us required a little improvement of I. M. Vinogradov’s estimations, getting by R. C. Vaughan.