Limit theorems in the space of analytic functions for the Hurwitz zeta-function with an algebraic irrational parameter

2011 ◽  
Vol 9 (2) ◽  
pp. 319-327
Author(s):  
Antanas Laurinčikas ◽  
Jörn Steuding
Keyword(s):  
Zeta Function ◽  
Analytic Functions ◽  
Limit Theorems ◽  
Hurwitz Zeta Function ◽  
Space Of Analytic Functions ◽  
Irrational Parameter

ON THE UNIVERSALITY OF THE HURWITZ ZETA-FUNCTION

2012 ◽  
Vol 09 (01) ◽  
pp. 155-165 ◽  
Author(s):  
ANTANAS LAURINČIKAS
Keyword(s):  
Analytic Function ◽  
Uniform Convergence ◽  
Zeta Function ◽  
Analytic Functions ◽  
Hurwitz Zeta Function ◽  
Space Of Analytic Functions ◽  
Rational Parameter ◽  
Compact Subsets

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.

scholarly journals ON APPROXIMATION OF ANALYTIC FUNCTIONS BY PERIODIC HURWITZ ZETA-FUNCTIONS

2018 ◽  
Vol 24 (1) ◽  
pp. 20-33 ◽  
Author(s):  
Darius Siaučiūnas ◽  
Violeta Franckevič ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Sequence ◽  
Hurwitz Zeta Function ◽  
Complex Numbers ◽  
Closed Set

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

scholarly journals Approximation of Analytic Functions by Shifts of Certain Compositions

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2583
Author(s):  
Darius Šiaučiūnas ◽  
Raivydas Šimėnas ◽  
Monika Tekorė
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Multidimensional Space ◽  
Space Of Analytic Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function.

scholarly journals New Result of Analytic Functions Related to Hurwitz Zeta Function

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
F. Ghanim ◽  
M. Darus
Keyword(s):  
Analytic Function ◽  
Linear Operator ◽  
Zeta Function ◽  
Analytic Functions ◽  
Hadamard Product ◽  
Hurwitz Zeta Function

By using a linear operator, we obtain some new results for a normalized analytic functionfdefined by means of the Hadamard product of Hurwitz zeta function. A class related to this function will be introduced and the properties will be discussed.

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