The joint universality for pairs of zeta functions in the Selberg class

2017 ◽  
Vol 151 (2) ◽  
pp. 282-327
Author(s):  
H. Mishou ◽  
H. Nagoshi
Keyword(s):  
Zeta Functions ◽  
Selberg Class ◽  
Joint Universality

scholarly journals The joint universality and the functional independence for Lerch zeta-functions

2000 ◽  
Vol 157 ◽  
pp. 211-227 ◽  
Author(s):  
Antanas Laurinčikas ◽  
Kohji Matsumoto
Keyword(s):  
Zeta Functions ◽  
Rational Numbers ◽  
Functional Independence ◽  
Joint Universality ◽  
Universality Theorem ◽  
Transcendental Numbers ◽  
Joint Universality Theorem

The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

ON INTERPOLATION FUNCTIONS FOR GENERALIZED Li COEFFICIENTS IN THE SELBERG CLASS

2011 ◽  
Vol 07 (03) ◽  
pp. 771-792
Author(s):  
ALMASA ODŽAK ◽  
LEJLA SMAJLOVIĆ
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Complex Function ◽  
Zeta Functions ◽  
Unitary Representations ◽  
Selberg Class ◽  
Interpolation Function ◽  
Entire Complex ◽  
Class Of Functions ◽  
Interpolation Functions

We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class [Formula: see text] that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GL n(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function [Formula: see text].

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.

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.

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 JOINT UNIVERSALITY OF DIRICHLET L-FUNCTIONS AND PERIODIC HURWITZ ZETA-FUNCTIONS

2012 ◽  
Vol 17 (5) ◽  
pp. 673-685 ◽  
Author(s):  
Kęstutis Janulis ◽  
Antanas Laurinčikas ◽  
Renata Macaitienė ◽  
Darius Šiaučiūnas
Keyword(s):  
Analytic Functions ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Joint Universality ◽  
Compact Subsets

In the paper, we prove that every system of analytic functions can be approximated simultaneously uniformly on compact subsets of some region by a collection consisting of shifts of Dirichlet L-functions with pairwise non-equivalent characters and periodic Hurwitz zeta-functions with parameters algebraically independent over the field of rational numbers.

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