Search

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z (s + i τ) , τ ∈ R , of the modified Mellin transform Z (s) of the square of the Riemann zeta-function ζ (1 / 2 + i t) . More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z (s + i τ) , which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure. [ABSTRACT FROM AUTHOR]

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ (s + i t k α 1) , ... , ζ (s + i t k α r)) of the Riemann zeta function. Here, { t k : k ∈ N } is the sequence of Gram numbers, and α 1 , ... , α r are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval [ N , N + M ] , here M = o (N) as N → ∞ , has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied. [ABSTRACT FROM AUTHOR]

Let t τ 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 / 2 Γ (s / 2) along the segment connecting points s = 1 / 2 and s = 1 / 2 + i t . t τ 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 + i t τ α 1) , ... , ζ (s + i t τ α r)) , where α 1 , ... , α r 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. [ABSTRACT FROM AUTHOR]