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

Abstract

Let tτ be a solution to the equation θ(t)=(τ−1)π, τ>0, where θ(t) is the increment of the argument of the function π−s/2Γ(s/2) along the segment connecting points s=1/2 and s=1/2+it. tτ is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function (ζ(s+itτα1),…,ζ(s+itταr)), where α1,…,αr are different positive numbers, in the interval [T,T+H] with H=o(T), T→∞, 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 ζ(s). Finally, an example of the approximation of analytic functions by a composition of the above shifts is given.