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 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 Joint approximation of analytic functions by shifts of the Riemann and periodic Hurwitz zeta-functions

2018 ◽  
Vol 12 (2) ◽  
pp. 508-527 ◽  
Author(s):  
Antanas Laurincikas ◽  
Renata Macaitienė
Keyword(s):  
Analytic Functions ◽  
Simultaneous Approximation ◽  
Linear Independence ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Critical Strip ◽  
Joint Approximation ◽  
Compact Subsets

We present some new results on the simultaneous approximation with given accuracy, uniformly on compact subsets of the critical strip, of a collection of analytic functions by discrete shifts of the Riemann and periodic Hurwitz zeta-functions. We prove that the set of such shifts has a positive lower density. For this, we apply the linear independence over the field of rational numbers of certain sets related to the zeta-functions.

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 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 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 JOINT DISCRETE APPROXIMATION OF A PAIR OF ANALYTIC FUNCTIONS BY PERIODIC ZETA-FUNCTIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 71-87 ◽  
Author(s):  
Aidas Balčiūnas ◽  
Virginija Garbaliauskienė ◽  
Julija Karaliūnaitė ◽  
Renata Macaitienė ◽  
Jurgita Petuškinaitė ◽  
...  
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Simultaneous Approximation ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Approximation Theorems ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.

The joint distribution of periodic zeta-functions

2011 ◽  
Vol 48 (2) ◽  
pp. 257-279 ◽  
Author(s):  
Roma kačinskaitė ◽  
Antanas Laurinčikas
Keyword(s):  
Analytic Functions ◽  
Joint Distribution ◽  
Zeta Functions ◽  
Functional Independence ◽  
Periodic Coefficients ◽  
Joint Approximation

In this paper, the joint approximation of a given collection of analytic functions by a collection of shifts of zeta-functions with periodic coefficients is obtained. This is applied to prove the functional independence for these zeta-functions.

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