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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 + 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 the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C 2 defined by means of the Epstein ζ (s ; Q) and Hurwitz ζ (s , α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]

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 this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip { s ∈ C : 1 / 2 < σ < 1 } . [ABSTRACT FROM AUTHOR]

In the paper, we consider the approximation of analytic functions by shifts from the wide class S ˜ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function L (s) ∈ S ˜ . Using the linear independence over Q of the multiset (h j log p : p ∈ P) , j = 1 , ... , r ; 2 π for positive h j , we obtain that there are many infinite shifts L (s + i k h 1) , ... , L (s + i k h r) , k = 0 , 1 , ... , approximating every collection f 1 (s) , ... , f r (s) of analytic non-vanishing functions defined in the strip { s ∈ C : σ L < σ < 1 } , where σ L is a degree of the function L (s) . For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. [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]

Suppose that Q is a positive defined n × n matrix, and Q [ x ̲ ] = x ̲ T Q x ̲ with x ̲ ∈ Z n . The Epstein zeta-function ζ (s ; Q) , s = σ + i t , is defined, for σ > n 2 , by the series ζ (s ; Q) = ∑ x ̲ ∈ Z n ∖ { 0 ̲ } (Q [ x ̲ ]) − s , and it has a meromorphic continuation to the whole complex plane. Let n ⩾ 4 be even, while φ (t) is an increasing differentiable function with a continuous monotonic bounded derivative φ ′ (t) such that φ (2 t) (φ ′ (t)) − 1 ≪ t , and the sequence { a φ (k) } is uniformly distributed modulo 1. In the paper, it is obtained that 1 N # N ⩽ k ⩽ 2 N : ζ (σ + i φ (k) ; Q) ∈ A , A ∈ B (C) , for σ > n − 1 2 , converges weakly to an explicitly given probability measure on (C , B (C)) as N → ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given. [ABSTRACT FROM AUTHOR]

The famous Selberg class is defined axiomatically and consists of Dirichlet series satisfying four axioms (Ramanujan hypothesis, analytic continuation, functional equation, multiplicativity). The Selberg–Steuding class S is a complemented Selberg class by an arithmetic hypothesis related to the distribution of prime numbers. In this paper, a joint universality theorem for the functions L from the class S on the approximation of a collection of analytic functions by shifts L (s + i a 1 τ) , ... , L (s + i a r τ) , where a 1 , ... , a r are real algebraic numbers linearly independent over the field of rational numbers, is obtained. It is proved that the set of the above approximating shifts is infinite, its lower density and, with some exception, density are positive. For the proof, a probabilistic method based on weak convergence of probability measures in the space of analytic functions is applied together with the Backer theorem on linear forms of logarithms and the Mergelyan theorem on approximation of analytic functions by polynomials. [ABSTRACT FROM AUTHOR]

Let θ (t) denote the increment of the argument of the product π − s / 2 Γ (s / 2) along the segment connecting the points s = 1 / 2 and s = 1 / 2 + i t , and t n denote the solution of the equation θ (t) = (n − 1) π , n = 0 , 1 , ... . The numbers t n 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 + i t k α 1) , ... , ζ (s + i t k α r)) , k = 0 , 1 , ... , where α 1 , ... , α r 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. [ABSTRACT FROM AUTHOR]

In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions. [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]

Let Q be a positive defined n × n matrix and Q [ x ̲ ] = x ̲ T Q x ̲ . The Epstein zeta-function ζ (s ; Q) , s = σ + i t , is defined, for σ > n 2 , by the series ζ (s ; Q) = ∑ x ̲ ∈ Z n \ { 0 ̲ } (Q [ x ̲ ]) − s , and is meromorphically continued on the whole complex plane. Suppose that n ⩾ 4 is even and φ (t) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1 T meas t ∈ [ 0 , T ] : ζ (σ + i φ (t) ; Q) ∈ A , A ∈ B (C) , converges weakly to an explicitly given probability measure on (C , B (C)) as T → ∞ . [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]

In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter. [ABSTRACT FROM AUTHOR]

Let 0 < γ1< γ2< ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ ℕ 0 } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied. [ABSTRACT FROM AUTHOR]

It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iт), т ∊ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if т take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when т takes values from the arithmetic progression fkh : k ∊ Ng with arbitrary fixed h > 0. For this, two types of h are considered. [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]

The Lerch zeta-function L(λ, α, s) with a transcendental parameter α, or with rational parameters α and λ is universal, i.e., a wide class of analytic functions is approximated by shifts L(λ, α, s + iτ), τ ∈ R. The case of an algebraic irrational α is an open problem. In the paper, it is proved that for all parameters α, 0 < α < 1, and λ, 0 < λ 6 1, including an algebraic irrational α, there exists a closed non-empty set of analytic functions Fα,λ such that every function f ∈ Fα,λ can be approximated by shifts L(λ, α, s + iτ). [ABSTRACT FROM AUTHOR]

We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s). [ABSTRACT FROM AUTHOR]

We consider a collection ofL-functions of elliptic curves twisted by a Dirichlet character moduloq(qis a prime number), and prove for this collection a joint limit theorem for weakly convergent probability measures in the space of analytic functions asq→ ∞. The limit measure is given explicitly. [ABSTRACT FROM PUBLISHER]

An universality theorem on the approximation of analytic functions by shifts ζ(s+iτ,F) of zeta-functions of normalized Hecke-eigen forms F, where τ takes values from the set {kαh:k=0,1,2,...} with fixed 0 < α < 1 and h > 0, is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove a multidimensional limit theorem for moduli of twists ofL-functions of elliptic curves. The limit measure in this theorem is defined by the characteristic transforms. [ABSTRACT FROM PUBLISHER]

The distribution of the vector (|ζ( s, α)|; ζ( s, α)), where ζ( s, α) is the Hurwitz zeta-function with transcendental parameter α, is considered and a probabilistic limit theorem is obtained. Also, the dependence between |ζ( s, α)| and ζ( s, α) in terms of m-characteristic transforms is discussed. [ABSTRACT FROM AUTHOR]

A limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function is obtained. [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]