Limit Theorems for the Riemann Zeta-Function in the Space of Analytic Functions

1996 ◽  
pp. 179-202 ◽  
Author(s):  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Limit Theorems ◽  
Space Of Analytic Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 ON DISCRETE UNIVERSALITY OF COMPOSITE FUNCTIONS

2012 ◽  
Vol 17 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Jovita Rašytė
Keyword(s):  
Analytic Function ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Space Of Analytic Functions ◽  
Composite Functions ◽  
The Riemann Zeta Function ◽  
Fixed Positive Number ◽  
Riemann Zeta ◽  
Discrete Universality

In 1975, S.M. Voronin proved that the Riemann zeta-function ζ (s) is universal in the sense that its shifts approximate uniformly on some sets any analytic function. Let h be a fixed positive number such that exp is irrational for all . In the paper, the classes of functions F such that the shifts F (ζ (s + imh)), , approximate any analytic function are presented. For the proof of theorems, some elements of the space of analytic functions are applied.

scholarly journals A Two-Dimentional Discrete Limit Theorem in the Space of Analytic Functions for Mellin Transforms of the Riemann Zeta-Function

2008 ◽  
Vol 13 (2) ◽  
pp. 159-167 ◽  
Author(s):  
V. Balinskaitė ◽  
V. Laurinčikas
Keyword(s):  
Limit Theorem ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Critical Line ◽  
Space Of Analytic Functions ◽  
Mellin Transforms ◽  
Convergence Of Probability Measures ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In the paper, a two-dimentional discrete limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for Mellin transforms of the Riemann zeta-function on the critical line 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 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 Some local-convexity theorems for the zeta-function-like analytic functions

1988 ◽  
Vol Volume 11 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Lower Bounds ◽  
Exponential Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Local Convexity ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience In this paper we investigate lower bounds for $$I(\sigma)= \int^H_{-H}\vert f(\sigma+it_0+iv)\vert^kdv,$$ where $f(s)$ is analytic for $s=\sigma+it$ in $\mathcal{R}=\{a\leq\sigma\leq b, t_0-H\leq t\leq t_0+H\}$ with $\vert f(s)\vert\leq M$ for $s\in\mathcal{R}$. Our method rests on a convexity technique, involving averaging with the exponential function. We prove a general lower bound result for $I(\sigma)$ and give an application concerning the Riemann zeta-function $\zeta(s)$. We also use our methods to prove that large values of $\vert\zeta(s)\vert$ are ``rare'' in a certain sense.

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