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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 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]

In the paper, it is obtained that different discrete shifts of Hurwitz zeta-functions simultaneously approximate a given collection of analytic functions. For this, a certain linear 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]

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]

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]