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By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ (s) by shifts ζ (s + i τ) . In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ (s) with all but at most countably many accuracies ε > 0 . Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. [ABSTRACT FROM AUTHOR]

In this article, we consider the asymptotic behaviour of the modified Mellin transform Z(s){\mathcal{Z}}\left(s), s=σ+its=\sigma +it, of the Riemann zeta-function using weak convergence of probability measures in the space of analytic functions defined by means of shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )), where φ(τ)\varphi \left(\tau ) is a real increasing to +∞+\infty differentiable function with monotonically decreasing derivative satisfying a certain estimate connected to the second moment of Z(s){\mathcal{Z}}\left(s). We prove in this case that the limit measure is concentrated at the point g0(s)≡0{g}_{0}\left(s)\equiv 0. This result is applied to the approximation of g0(s){g}_{0}\left(s) by shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )).

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ(s) by shifts ζ(s+iτ). In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ(s) with all but at most countably many accuracies ε>0. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.

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]

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N + M] with with real algebraic numbers a1,...,ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.

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, a theorem on approximation of a wide class of analytic functions by generalized shifts uT (s + if(t )) of an absolutely convergent Dirichlet series uT (s) which in the mean is close to the Riemann zetafunction is obtained. Here f(t) is a monotonically increasing differentiable function having a monotonic continuous derivative such that f(2t) max tt2t 1 f'(t) « t as t 8, and uT 8 and uT « T2 as T 8. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z(s+iτ), τ∈R, of the modified Mellin transform Z(s) of the square of the Riemann zeta-function ζ(1/2+it). More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z(s+iτ), 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.

In the paper, it is obtained that there are infinite discrete shifts Ξ(s+ikh), h>0, k∈N0 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.

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]

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ(s+itkα1),…,ζ(s+itkαr)) of the Riemann zeta function. Here, {tk:k∈N} is the sequence of Gram numbers, and α1,…,αr 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.

In 2007, H. Mishou proved the universality theorem on the joint approximation of a pair of analytic functions by the shifts of the Riemann zeta-function and the Hurwitz zeta-function with transcendental parameter . In this paper, we obtain a similar theorem on approximation by the shifts , , , where and are absolutely convergent Dirichlet series, and, as , they tend in mean to and respectively. [ABSTRACT FROM AUTHOR]

By the Mishou theorem (Mishou in Lith Math J 47:32–47, 2007), the set of shifts (ζ (s + i τ) , ζ (s + i τ , α)) , τ ∈ R , of the Riemann zeta-function and Hurwitz zeta-function with transcendental parameter simultaneously approximating a pair of analytic functions has a positive lower density. In the paper, two absolutely convergent Dirichlet series close to the Riemann and Hurwitz zeta-functions are constructed, and it is proved that the set of their shifts approximating a pair of analytic functions has a positive density. For the proof, probabilistic arguments are applied. [ABSTRACT FROM AUTHOR]

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.

In the paper, the simultaneous approximation of a tuple of analytic functions in the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) of the Riemann zeta-function ζ(s) with a certain class of continuously differentiable increasing functions φ1,…,φr is considered. This class of functions φ1,…,φr 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 (φj(τ))1/3(logφj(τ))26/15 and derivatives φj′(τ). The density problem is also discussed. An example of the approximation by a composition F(ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) with a certain continuous operator F in the space of analytic functions is given.