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The Hurwitz zeta-function ζ (s , α) , s = σ + i t , with parameter 0 < α ⩽ 1 is a generalization of the Riemann zeta-function ζ (s) ( ζ (s , 1) = ζ (s) ) and was introduced at the end of the 19th century. The function ζ (s , α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ (s , α) is the main example of zeta-functions without Euler's product (except for the cases α = 1 , α = 1 / 2 ), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ (s , α) for some α is universal, i.e., its shifts ζ (s + i τ , α) , τ ∈ R , approximate every analytic function defined in the strip { s : 1 / 2 < σ < 1 } . For needs of effectivization of the universality property for ζ (s , α) , the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ (σ + i t , α) in short intervals. In this paper, we obtain the bound O (H) for that mean square over the interval [ T − H , T + H ] , with T 27 / 82 ⩽ H ⩽ T σ and 1 / 2 < σ ⩽ 7 / 12 . This is the first result on the mean square for ζ (s , α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ (s , α) and other zeta-functions in short intervals. [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]

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed. [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 this paper, the approximation of analytic functions by shifts  ζ P (s + i τ)  of Beurling zeta-functions  ζ P (s)  of certain systems  P  of generalized prime numbers is discussed. It is required that the system of generalized integers  N P  generated by  P  satisfies  ∑ m ⩽ x , m ∈ N 1 = a x + O (x δ) ,  a > 0 ,  0 ⩽ δ < 1 , and the function  ζ P (s)  in some strip lying in  σ ^ < σ < 1 ,  σ ^ > δ , which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. [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, the approximation of analytic functions on compact sets of the strip { s = σ + i t ∈ C ∣ 1 / 2 < σ < 1 } by shifts F (ζ (s + i u 1 (τ)) , ... , ζ (s + i u r (τ))) , where ζ (s) is the Riemann zeta-function, u 1 , ... , u r are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval [ T , T + H ] with H = o (T) , T → ∞ , has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given. [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]

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 consider the simultaneous approximation of a collection of analytic functions by a collection of shifts of the Riemann zeta-function (ζ(s + itα1τ ), . . ., ζ(s + itαrτ )), where tτ is the Gram function and α1, . . ., αr are different positive numbers. It is obtained that the set of such shifts has a positive lower density. [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]

We consider one absolutely convergent series... connected to the Riemann zeta-function, and prove a theorem on joint approximation of analytic functions by shifts... as... are fixed different positive numbers... is the Gram function, and... [ABSTRACT FROM AUTHOR]

In the paper, a theorem on the approximation of collections of a wide class of analytic functions by collections of shifts of zeta-functions with periodic co-efficients involving imaginary parts of non-trivial zeros of the Riemann zeta-function is obtained. For this, a version of the Montgomery pair correlation conjecture is required. [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]

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]

In the paper, a joint theorem on the approximation of collections of analytic functions by generalized discrete shifts of zeta-functions with periodic coefficients is obtained. The latter result extend theorems of [9]. [ABSTRACT FROM AUTHOR]

In 2007, H. Mishou obtained a joint universality theorem for the Riemann and Hurwitz zeta-functions ζ(s) and ζ(s; α) with transcendental parameter α on the approximation of a pair of analytic functions by shifts (ζ(s+i); ζ(s+i; α)), 2 R. In the paper, the Mishou theorem is generalized for the set of above shifts having a weighted positive lower density. Also, the case of a positive density is considered. [ABSTRACT FROM AUTHOR]

In the paper, for certain classes of operators F in the space of analytic functions, we prove the discrete universality for compositions F (ζ(s, α; , )), where ζ(s, α; ; ) is a collection consisting from periodic and periodic Hurwitz zeta-functions, i. e., the approximation of analytic functions by discrete shifts F (ζ(s + ikh, α; ; )) with h > 0 and k = 0, 1,.... For this, a theorem of [12] is applied. [ABSTRACT FROM AUTHOR]

The paper is a continuation of Korolev and Laurinčikas (Aequ Math 93:859–873, 2019), where theorems on the approximation of analytic functions by shifts ζ (s + i h t k) , h > 0 , k ∈ N , and t k are the Gram points, were obtained. In this paper, it is proved, that the set of shifts ζ (s + i h t k) has a positive density in short intervals [ N , N + M ] with M = o (N) . [ABSTRACT FROM AUTHOR]

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ζ(s+ikh) with fixed h > 0 when positive integers k run over the interval [N,N+M], where N1/3(logN)26=15 ≤ M ≤ N, and prove that those k have a positive lower density as N → ∞. The same is true for some compositions. Two types of h > 0 are discussed separately. [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 Gram points t n are defined as solutions of the equation θ (t) = (n - 1) π , n ∈ N , where θ (t) , t > 0 , denotes the increment of the argument of the function π - s / 2 Γ s 2 along the segment connecting the points s = 1 2 and s = 1 2 + i t . In the paper, theorems on the approximation of a wide class of analytic functions by shifts ζ (s + i h t k) , h > 0 , k ∈ N , of the Riemann zeta-function are obtained. [ABSTRACT FROM AUTHOR]

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied. [ABSTRACT FROM AUTHOR]

The classical Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts of the Riemann zeta-function ζ(s + iτ), τ ∈ ℝ. In the paper, we generalize this approximation for shifts ζ(s + iφ(τ)), where 𝜑(τ) has the monotonic positive derivative such that 1/𝜑′(τ) = o(τ) and φ(2τ) × maxτ ≪ t ≪ 2τ(1/φ′(t)) ≪ τ as τ →∞. [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]

The periodic Hurwitz zeta-function (s, α a), s = σ + it, is a generalization of the classical Hurwitz zeta-function, and is defined, for σ > 1, by the series (s, α a) = ... where a = {am} is a periodic sequence of complex numbers, and 0 < α ≤ 1 is a fixed parameter. In the paper, theorems on the approximation of analytic functions by discrete shifts (s + ikh, α a), k = 0, 1, . . ., h > 0, with rational α are obtained. [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]

In the paper, a discrete universality theorem for the Hurwitz zeta-function ζ ( s , α ) on the approximation of analytic functions by shifts ζ ( s + i τ , α ) when τ takes values from the set { k β h : k = 0 , 1 , … } with fixed 0 < β < 1 and h > 0 is obtained. For its proof, the uniform distribution modulo 1 is applied. [ABSTRACT FROM AUTHOR]

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]

We prove a discrete version of the Mishou theorem on the approximation of every pair of analytic functions by shifts of the Riemann zeta-function and Hurwitz zeta-functions. For this, a certain independence hypothesis is applied. [ABSTRACT FROM AUTHOR]

In the paper, a discrete universality theorem on the approximation of analytic functions by discrete shifts of the Hurwitz zeta-function is proved. The parameter of the Hurwitz zeta-function and the step of discrete shifts are related by a certain linear independence relation. Also, discrete universality theorems for composite functions are discussed. [ABSTRACT FROM AUTHOR]

It is known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that its shifts ζ(s + iτ, α), τ ∈ ℝ, approximate with a given accuracy any analytic function uniformly on compact subsets of the strip D = {s ∈ ℂ : ½ < σ < 1}. Let H(D) denote the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In the paper, the classes of functions F : H(D) → H(D) such that F(ζ(s, α)) is universal in the above sense are considered. For example, if F is continuous and, for each polynomial p = p(s), the set F-1{p} is non-empty, then F(ζ(s, α)) with transcendental α is universal. [ABSTRACT FROM AUTHOR]

Abstract: We construct classes of composite functions of the Riemann zeta-function and Hurwitz zeta function with transcendental parameter which are universal in the sense that their shifts uniformly on compact subsets of some region approximate any analytic function. For example, the functions , and are universal. [Copyright &y& Elsevier]

It is well known that Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers are universal in the sense that their shifts approximate simultaneously any collection of analytic functions. In this paper we introduce some classes of universal composite functions of a collection of Hurwitz zeta-functions. [ABSTRACT FROM AUTHOR]

Abstract: In 1975, S.M. Voronin proved the universality of the Riemann zeta-function . This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts . In the paper, we consider the functions which are universal in the Voronin sense. [Copyright &y& Elsevier]

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]

Let 0 < γ 1 < γ 2 < ⋯ ⩽ γ k ⩽ ⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ (s) . Using a certain estimate on the pair correlation of the sequence { γ k } in the intervals [ N , N + M ] with N 1 / 2 + ε ⩽ M ⩽ N , we prove that the set of shifts ζ (s + i h γ k) , h > 0 , approximating any non-vanishing analytic function defined in the strip { s ∈ C : 1 / 2 < Re s < 1 } with accuracy ε > 0 has a positive lower density in [ N , N + M ] as N → ∞ . Moreover, this set has a positive density for all but at most countably ε > 0 . The above approximation property remains valid for certain compositions F (ζ (s)) . [ABSTRACT FROM AUTHOR]

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions L (s + i a 1 t τ , χ 1) , ... , L (s + i a r t τ , χ r) , where a 1 , ... , a r are non-zero real algebraic numbers linearly independent over the field Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density. [ABSTRACT FROM AUTHOR]