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In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . ., ζ(s + 2πiark)), k ∈ N, approximating analytic non-vanishing functions f1(s), . . ., fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . ., ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series. [ABSTRACT FROM AUTHOR]

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 the paper, it is obtained that there are infinite discrete shifts Ξ (s + i k h) , h > 0 , k ∈ N 0 of the Mellin transform Ξ (s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. [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]

In the paper, the simultaneous approximation of a tuple of analytic functions in the strip { s = σ + i t ∈ C : 1 / 2 < σ < 1 } by shifts (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) of the Riemann zeta-function ζ (s) with a certain class of continuously differentiable increasing functions φ 1 , ... , φ r is considered. This class of functions φ 1 , ... , φ r is characterized by the growth of their derivatives. It is proved that the set of mentioned shifts in the interval [ T , T + H ] with H = o (T) has a positive lower density. The precise expression for H is described by the functions (φ j (τ)) 1 / 3 (log φ j (τ)) 26 / 15 and derivatives φ j ′ (τ) . The density problem is also discussed. An example of the approximation by a composition F (ζ (s + i φ 1 (τ)) , ... , ζ (s + i φ r (τ))) with a certain continuous operator F in the space of analytic functions is given. [ABSTRACT FROM AUTHOR]

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ (s + i τ) , s = σ + i t , τ ∈ R , that approximate any given non-vanishing analytic function defined on { s ∈ C : 1 2 < σ < 1 } has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [ T , T + H ] with T 1 / 3 (log ⁡ T) 26 / 15 ⩽ H ⩽ T. [ABSTRACT FROM AUTHOR]

In the paper, a Dirichlet series ζ u N (s) whose shifts ζ u N (s + i k h) , k = 0 , 1 , ⋯ , h > 0 , approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence u N → ∞ and u N ≪ N 2 as N → ∞ . [ABSTRACT FROM AUTHOR]