Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II

Let t(tau) be a solution to the equation theta (t) = (tau - 1)pi, tau > 0, where theta (t) is the increment of the argument of the function pi(-s/2)Gamma (s/2) along the segment connecting points s = 1 / 2 and s = 1/2 + it. t(tau) is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function (zeta(s + it(tau)(alpha 1)), ..., zeta (s + it(tau)(alpha r))), where alpha(1), ..., alpha(r) are different positive numbers, in the interval [T, T + H] with H = o (T), T -> infinity, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to zeta(s). Finally, an example of the approximation of analytic functions by a composition of the above shifts is given.