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Let a = { a m : m ∈ N } be a periodic multiplicative sequence of complex numbers and L (s ; a) , s = σ + i t a Dirichlet series with coefficients a m . In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1 / 2 < σ < 1 via discrete shifts L (s + i h t k ; a) , h > 0 , k ∈ N , where { t k : k ∈ N } is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied. [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]

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]

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]

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]