scholarly journals Approximation by Shifts of Compositions of Dirichlet L-Functions with the Gram Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 751
Author(s):  
Artūras Dubickas ◽  
Ramūnas Garunkštis ◽  
Antanas Laurinčikas
Keyword(s):  
Analytic Functions ◽  
Algebraic Numbers ◽  
Linearly Independent ◽  
Joint Approximation

In this paper, a joint approximation of analytic functions by shifts of Dirichlet L-functions L ( s + i a 1 t τ , χ 1 ) , … , L ( s + i a r t τ , χ r ) , where a 1 , … , a r are non-zero real algebraic numbers linearly independent over the field Q and t τ is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.

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 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 Further Variations on the Six Exponentials Theorem.

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Present Note ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Linearly Independent ◽  
International Audience ◽  
Rank 2

International audience According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $$ \lambda_{21}\lambda_{12}-\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}-\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}-\lambda_{13}\lambda_{21} $$ is transcendental.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

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.

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.

Distributional boundary values of analytic functions and positive definite distributions

2015 ◽  
Vol 20 (2) ◽  
pp. 210-225
Author(s):  
Saulius Norvidas ◽  
Keyword(s):  
Analytic Functions ◽  
Positive Definite ◽  
Boundary Values

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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