scholarly journals Extension of the discrete universality theorem for zeta-functions of certain cusp forms

2018 ◽  
Vol 23 (6) ◽  
pp. 961-973
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas ◽  
Adelė Vaiginytė
Keyword(s):  
Analytic Functions ◽  
Zeta Functions ◽  
Growth Conditions ◽  
Cusp Forms ◽  
Continuous Derivative ◽  
Natural Growth ◽  
Universality Theorem ◽  
Discrete Universality ◽  
Positive Integers

In the paper, an universality theorem on the approximation of analytic functions by generalized discrete shifts of zeta functions of Hecke-eigen cusp forms is obtained. These shifts are defined by using the function having continuous derivative satisfying certain natural growth conditions and, on positive integers, uniformly distributed modulo 1.

scholarly journals On joint approximation of analytic functions by nonlinear shifts of zeta-functions of certain cusp forms

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas ◽  
Adelė Vaiginytė
Keyword(s):  
Analytic Functions ◽  
Simultaneous Approximation ◽  
Zeta Functions ◽  
Growth Conditions ◽  
Cusp Forms ◽  
Differentiable Functions ◽  
Joint Approximation ◽  
Discrete Universality ◽  
Positive Integers

In the paper, joint discrete universality theorems on the simultaneous approximation of a collection of analytic functions by a collection of discrete shifts of zeta-functions attached to normalized Hecke-eigen cusp forms are obtained. These shifts are defined by means of nonlinear differentiable functions that satisfy certain growth conditions, and their combination on positive integers is uniformly distributed modulo 1.

scholarly journals Uniform distribution modulo 1 and the universality of zeta-functions of certain cusp forms

2016 ◽  
Vol 100 (114) ◽  
pp. 131-140
Author(s):  
Antanas Laurincikas
Keyword(s):  
Uniform Distribution ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Cusp Forms ◽  
Distribution Modulo 1 ◽  
Uniform Distribution Modulo 1 ◽  
Universality Theorem ◽  
Distribution Modulo

An universality theorem on the approximation of analytic functions by shifts ?(s+i?,F) of zeta-functions of normalized Hecke-eigen forms F, where ? takes values from the set {k?h:k=0,1,2,...} with fixed 0 < ? < 1 and h > 0, is obtained.

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 &lt; α &lt; 1, and h &gt; 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 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 A WEIGHTED VERSION OF THE MISHOU THEOREM

2021 ◽  
Vol 26 (1) ◽  
pp. 21-33
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas ◽  
Gediminas Vadeikis
Keyword(s):  
Analytic Functions ◽  
Zeta Functions ◽  
Positive Density ◽  
Weighted Version ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

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+iτ),ζ(s+iτ,α)), τ 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.

scholarly journals Discrete universality of the Riemann zeta-function in short intervals

2020 ◽  
pp. 19-19
Author(s):  
Antanas Laurincikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality ◽  
Positive Integers

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ?(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.

scholarly journals Joint Universality of the Zeta-Functions of Cusp Forms

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2161
Author(s):  
Renata Macaitienė
Keyword(s):  
Cusp Form ◽  
Modular Group ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Cusp Forms ◽  
Algebraic Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Linearly Independent ◽  
Joint Universality Theorem

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),⋯,ζ(s+ihrτ,F)) is proved. Here, h1,⋯,hr are algebraic numbers linearly independent over the field of rational numbers.

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.

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