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 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.

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.

The Riemann hypothesis and universality of the Riemann zeta-function

2018 ◽  
Vol 68 (4) ◽  
pp. 741-748 ◽  
Author(s):  
Ramūnas Garunkštis ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Riemann Hypothesis ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).

Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function

2020 ◽  
Vol 57 (2) ◽  
pp. 147-164
Author(s):  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
Weak Form ◽  
Rational Numbers ◽  
Hurwitz Zeta Function ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Universality

AbstractLet 0 < γ1 < γ2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + iγkh, α), h > 0, with parameter α such that the set {log(m + α): m ∈ } is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γk} is applied.

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.

scholarly journals HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

2020 ◽  
Vol 71 (1) ◽  
pp. 257-280
Author(s):  
Jeffrey C Lagarias ◽  
Brad Rodgers
Keyword(s):  
Correlation Function ◽  
Zeta Function ◽  
Point Process ◽  
Riemann Zeta Function ◽  
Alternative Hypothesis ◽  
Pair Correlation ◽  
Local Statistics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Rule Out

Abstract The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

On the Pair Correlation of Zeros of the Riemann Zeta-Function

2000 ◽  
Vol 80 (1) ◽  
pp. 31-49 ◽  
Author(s):  
D. A. Goldston ◽  
S. M. Gonek ◽  
A. E. Özlük ◽  
C. Snyder
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Pair Correlation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta
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