Series representing transcendental numbers that are not U-numbers

2015 ◽  
Vol 11 (03) ◽  
pp. 869-892
Author(s):  
Emre Alkan
Keyword(s):  
Rational Functions ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Type Series ◽  
Transcendental Numbers ◽  
Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals The number of exceptional approximations in Roth's theorem

1995 ◽  
Vol 59 (3) ◽  
pp. 375-383 ◽  
Author(s):  
Wolfgang M. Schmidt
Keyword(s):  
Algebraic Number ◽  
Rational Numbers ◽  
Upper Bounds

AbstractRoth's Theorem says that given ρ < 2 and an algebraic number α, all but finitely many rational numbers x/y satisfy |α - (x/y)|< |y|-ρ. We give upper bounds for the number of these exceptional rationals when 3 ≤ ρ ≤ d, where d is the degree of α. Our result suplements bounds given by Bombieri and Van der Poorten when 2 > ρ ≤ 3; naturally the bounds become smaller as ρ increases.

scholarly journals The Completed Finite Period Map and Galois Theory of Supercongruences

2018 ◽  
Vol 2019 (23) ◽  
pp. 7379-7405
Author(s):  
Julian Rosen
Keyword(s):  
Algebraic Number ◽  
Complex Number ◽  
Galois Theory ◽  
Rational Numbers ◽  
Multiple Zeta Values ◽  
Period Map ◽  
Transcendental Numbers ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Finite Period

Abstract A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.

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.

Parallel fluid queues with constant inflows and simultaneous random reductions

2001 ◽  
Vol 38 (3) ◽  
pp. 609-620 ◽  
Author(s):  
Offer Kella ◽  
Masakiyo Miyazawa
Keyword(s):  
Poisson Process ◽  
Upper Bounds ◽  
Decay Rates ◽  
Fluid Queue ◽  
Linear Combinations ◽  
Fluid Queues ◽  
Constant Rate ◽  
Major Interest ◽  
Dimensional Distribution ◽  
Queues In Parallel

We consider I fluid queues in parallel. Each fluid queue has a deterministic inflow with a constant rate. At a random instant subject to a Poisson process, random amounts of fluids are simultaneously reduced. The requested amounts for the reduction are subject to a general I-dimensional distribution. The queues with inventories that are smaller than the requests are emptied. Stochastic upper bounds are considered for the stationary distribution of the joint buffer contents. Our major interest is in finding exponential product-form bounds, which turn out to have the appropriate decay rates with respect to certain linear combinations of buffer contents.

Counting points of given height that generate a quadratic extension of a function field

2015 ◽  
Vol 11 (02) ◽  
pp. 569-592 ◽  
Author(s):  
David Kettlestrings ◽  
Jeffrey Lin Thunder
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Function Field ◽  
Asymptotic Estimate ◽  
Algebraic Closure ◽  
Quadratic Extension ◽  
Rational Functions ◽  
Rational Numbers ◽  
Algebraic Extension ◽  
Counting Points

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

scholarly journals Perfect Matchings for the Three-Term Gale-Robinson Sequences

10.37236/214 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
James Propp ◽  
Julian West
Keyword(s):  
Recurrence Relations ◽  
Initial Conditions ◽  
A Priori ◽  
Rational Functions ◽  
Rational Numbers ◽  
Perfect Matchings ◽  
Parameter Family ◽  
Integer Coefficients ◽  
Special Cases

In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the '90s, proofs of integrality were known only for individual special cases. In the early '00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson's integrality conjecture. They actually proved much more, and in particular, that certain bivariate rational functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients. However, their proof did not offer any enumerative interpretation of the Gale-Robinson numbers/polynomials. Here we provide such an interpretation in the setting of perfect matchings of graphs, which makes integrality/polynomiality obvious. Moreover, this interpretation implies that the coefficients of the Gale-Robinson polynomials are positive, as Fomin and Zelevinsky conjectured.

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