Mathematics, Vol. 11, Pages 565: Joint Discrete Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by Gram Points

Mathematics, Vol. 11, Pages 565: Joint Discrete Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by Gram Points

Mathematics doi: 10.3390/math11030565

Authors: Antanas Laurinčikas

Let θ(t) denote the increment of the argument of the product π−s/2Γ(s/2) along the segment connecting the points s=1/2 and s=1/2+it, and tn denote the solution of the equation θ(t)=(n−1)π, n=0,1,⋯. The numbers tn are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function (ζ(s+itkα1),⋯,ζ(s+itkαr)), k=0,1,⋯, where α1,⋯,αr are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied.