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.

scholarly journals Dirichlet summations and products over primes

1993 ◽  
Vol 16 (2) ◽  
pp. 359-372 ◽  
Author(s):  
Geoffrey B. Campbell
Keyword(s):  
Zeta Functions ◽  
Rational Numbers ◽  
Infinite Products ◽  
Riemann Zeta ◽  
Alpha Beta

We derive new classes of infinite products taken over the primes, for example expressing∏p(11−p−n)(1−p−m)−1as an infinite produce of Riemann zeta functions, this product being taken over the set of rational numbersα/βgeater than zero with a relatively prime toβζ(n)∏α,βζ(αm+βn)1/β.

Chowla’s theorem over function fields

2018 ◽  
Vol 14 (06) ◽  
pp. 1689-1698
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Dirichlet Series ◽  
Function Field ◽  
Function Fields ◽  
Rational Numbers ◽  
Periodic Coefficients ◽  
Linearly Independent ◽  
Parity Condition

Sarvadaman Chowla proved that if [Formula: see text] is an odd prime, then [Formula: see text] ([Formula: see text]) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker–Birch–Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at [Formula: see text] in the function field setup with a parity condition.

Singular n-tuples and Hausdorff dimension

1977 ◽  
Vol 81 (3) ◽  
pp. 377-385 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Measure Zero ◽  
Rational Numbers ◽  
Linearly Independent ◽  
Singular Numbers ◽  
Image Position ◽  
Almost All ◽  
Dimensional Lebesgue Measure

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We writeIf α is a real number, ∥α∥ denotes the distance from α to the nearest integer.Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).for all X ≥ 1. We say that θ is singular ifSingular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed thatfor almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

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 A Criterion for linear independence of infinite products

2015 ◽  
Vol 23 (2) ◽  
pp. 107-120
Author(s):  
Jaroslav Hančl ◽  
Ondřej Kolouch ◽  
Lukáš Novotný
Keyword(s):  
Linear Independence ◽  
Rational Numbers ◽  
Infinite Products

Abstract Using an idea of Erdős the paper establishes a criterion for the linear independence of infinite products which consist of rational numbers. A criterion for irrationality is obtained as a consequence.

Singular n-tuples and Hausdorff dimension. II

1992 ◽  
Vol 111 (3) ◽  
pp. 577-584 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Natural Number ◽  
Rational Numbers ◽  
Inner Product ◽  
Linearly Independent ◽  
Image Position

Let n be a natural number, with n ≥ 2. Let Kn denote the set of θ in Euclidean space Rn for which θ1, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y,…. We writeInner product is denoted by θø. In Rl, ‖θ‖ denotes distance to the nearest integer.

scholarly journals Efficient Algorithms for Creation of Linearly-independent Decision Diagrams and their Mapping to Regular Layouts

VLSI Design ◽  
2002 ◽  
Vol 14 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Marek Perkowski ◽  
Bogdan Falkowski ◽  
Malgorzata Chrzanowska-Jeske ◽  
Rolf Drechsler
Keyword(s):  
Decision Diagrams ◽  
Regular Structures ◽  
Linearly Independent ◽  
New Concepts ◽  
Starting Point ◽  
New Type ◽  
Multi Level ◽  
Input Variables ◽  
Independent Decision ◽  
Higher Regularity

A new kind of a decision diagrams are presented: its nodes correspond to all types of nonsingular expansions for groups of input variables, in particular pairs. The diagrams are called the Linearly Independent (LI) Decision Diagrams (LI DDs). There are 840 nonsigular expansions for a pair of variables, thus 840 different types of nodes in the tree. Therefore, the number of nodes in such (exact) diagrams is usually much smaller than the number of nodes in the well-known Kronecker diagrams (which have only single-variable Shannon, Positive Davio, and Negative Davio expansions in nodes). It is usually much smaller than 1/3 of the number of nodes in Kronecker diagrams. Similarly to Kronecker diagrams, the LI Diagrams are a starting point to a synthesis of multilevel AND/OR/EXOR circuits with regular structures. Other advantages of LI diagrams include: they generalize the well-known Pseudo-Kronecker Functional Decision Diagrams, and can be used to optimize the new type of PLAs called LI PLAs. Importantly, while the known decision diagrams used AND/EXOR or AND/OR bases, the new diagrams are AND/OR/EXOR-based. Thus, because of a larger design space, multi-level structures of higher regularity can be created with them. This paper presents both new concepts and new efficient synthesis algorithms.

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.

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.

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).

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