scholarly journals A WEIGHTED UNIVERSALITY THEOREM FOR PERIODIC ZETA-FUNCTIONS

2017 ◽  
Vol 22 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Renata Macaitienė ◽  
Mindaugas Stoncelis ◽  
Darius Šiaučiūnas
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Complex Plane ◽  
Wide Class ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Coefficients ◽  
Universality Theorem

The periodic zeta-function ζ(s; a), s = σ + it is defined for σ > 1 by the Dirichlet series with periodic coefficients and is meromorphically continued to the whole complex plane. It is known that the function ζ(s; a), for some sequences a of coefficients, is universal in the sense that its shifts ζ(s + iτ ; a), τ ∈ R, approximate a wide class of analytic functions. In the paper, a weighted universality theorem for the function ζ(s; a) is obtained.

scholarly journals Joint universality of periodic zeta-functions with multiplicative coefficients

2020 ◽  
Vol 25 (5) ◽  
Author(s):  
Antanas Laurinčikas ◽  
Monika Tekorė
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Coefficients ◽  
Joint Universality

The periodic zeta-function is defined by the ordinary Dirichlet series with periodic coefficients. In the paper, joint universality theorems on the approximation of a collection of analytic functions by nonlinear shifts of periodic zeta-functions with multiplicative coefficients are obtained. These theorems do not use any independence hypotheses on the coefficients of zeta-functions.

scholarly journals A WEIGHTED DISCRETE UNIVERSALITY THEOREM FOR PERIODIC ZETA-FUNCTIONS. II

2017 ◽  
Vol 22 (6) ◽  
pp. 750-762 ◽  
Author(s):  
Renata Macaitienė ◽  
Mindaugas Stoncelis ◽  
Darius Šiaučiūnas
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Sequence ◽  
Universality Theorem ◽  
Discrete Universality

In the paper, a weighted theorem on the approximation of a wide class of analytic functions by shifts ζ(s + ikαh; a), k ∈ N, 0 < α < 1, and h > 0, of the periodic zeta-function ζ(s; a) with multiplicative periodic sequence a, is obtained.

scholarly journals A MIXED JOINT UNIVERSALITY THEOREM FOR ZETA-FUNCTIONS. II

2014 ◽  
Vol 19 (1) ◽  
pp. 52-65 ◽  
Author(s):  
Vaida Pocevičienė ◽  
Darius Šiaučiūnas
Keyword(s):  
Zeta Function ◽  
Cusp Form ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem ◽  
Independent Parameters

In the paper, a joint universality theorem on the approximation of analytic functions for zeta-function of a normalized Hecke eigen cusp form and a collection of periodic Hurwitz zeta-functions with algebraically independent parameters is obtained.

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 UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

2021 ◽  
Vol 26 (1) ◽  
pp. 82-93
Author(s):  
Aidas Balčiūnas ◽  
Violeta Franckevič ◽  
Virginija Garbaliauskienė ◽  
Renata Macaitienė ◽  
Audronė Rimkevičienė
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Cusp Forms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

scholarly journals WEIGHTED DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION

2020 ◽  
Vol 25 (1) ◽  
pp. 21-36
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas ◽  
Gediminas Vadeikis
Keyword(s):  
Zeta Function ◽  
Arithmetic Progression ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Universality Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality

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

A MIXED JOINT UNIVERSALITY THEOREM FOR ZETA‐FUNCTIONS

2010 ◽  
Vol 15 (4) ◽  
pp. 431-446 ◽  
Author(s):  
Jonas Genys ◽  
Renata Macaitienė ◽  
Santa Račkauskienė ◽  
Darius Šiaučiūnas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Joint Universality ◽  
Universality Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Joint Universality Theorem

In the paper, a joint universality theorem for the Riemann zeta‐function and a collection of periodic Hurwitz zeta‐functions on approximation of analytic functions is obtained.

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

The joint universality for periodic Hurwitz zeta-functions

Analysis ◽  
2006 ◽  
Vol 26 (3) ◽  
Author(s):  
Antanas Laurinčikas
Keyword(s):  
Zeta Functions ◽  
Periodic Coefficients ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

We prove a joint universality theorem for the Hurwitz zeta-functions with periodic coefficients.

scholarly journals Joint universality of the Riemann zeta-function and Lerch zeta-functions

2013 ◽  
Vol 18 (3) ◽  
pp. 314-326
Author(s):  
Antanas Laurinčikas ◽  
Renata Macaitienė˙
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Joint Universality Theorem

In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch zeta-functions with parameters algebraically independent over the field of rational numbers.

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