Transcendental Numbers Research Articles

Ax–Schanuel type theorems on functional transcendence via Nevanlinna theory

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples to show that these results are optimal. As a byproduct, we also show that analytic dependence implies algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax–Schanuel theorem will be discussed.

Undecidable arithmetic properties of solutions of Fredholm integral equations

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f(α) is transcendental or algebraic if f(z) is an E-function and α∈Q‾⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f(z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f(0)∈Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6].

The Discrete Case of the Mixed Joint Universality for a Class of Certain Partial Zeta-functions

We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions $\varphi_h(s)$ and a collection of periodic Hurwitz zeta-functions $\zeta(s,\alpha;\mathfrak{B})$ under the condition that the common difference of arithmetical progression $h \gt 0$ is such that $\exp \big\{ \frac{2\pi}{h} \big\}$ is a rational number and parameter $\alpha$ is a transcendental number.

N3LO gravitational quadratic-in-spin interactions at G4

We compute the N3LO gravitational quadratic-in-spin interactions at G4 in the post-Newtonian (PN) expansion via the effective field theory (EFT) of gravitating spinning objects for the first time. This result contributes at the 5PN order for maximally-spinning compact objects, adding the spinning case to the static sector at this PN accuracy. This sector requires extending the EFT of a spinning particle beyond linear order in the curvature to include higher-order operators quadratic in the curvature that are relevant at this PN order. We make use of a diagrammatic expansion in the worldline picture, and rely on our recent upgrade of the EFTofPNG code, which we further extend to handle this sector. Similar to the spin-orbit sector, we find that the contributing three-loop graphs give rise to divergences, logarithms, and transcendental numbers. However, in this sector all of these features conspire to cancel out from the final result, which contains only finite rational terms.

N3LO gravitational spin-orbit coupling at order G4

In this paper we derive for the first time the N3LO gravitational spin-orbit coupling at order G4 in the post-Newtonian (PN) approximation within the effective field theory (EFT) of gravitating spinning objects. This represents the first computation in a spinning sector involving three-loop integration. We provide a comprehensive account of the topologies in the worldline picture for the computation at order G4. Our computation makes use of the publicly-available EFTofPNG code, which is extended using loop-integration techniques from particle amplitudes. We provide the results for each of the Feynman diagrams in this sector. The three-loop graphs in the worldline picture give rise to new features in the spinning sector, including divergent terms and logarithms from dimensional regularization, as well as transcendental numbers, all of which survive in the final result of the topologies at this order. This result enters at the 4.5PN order for maximally-rotating compact objects, and together with previous work in this line, paves the way for the completion of this PN accuracy.

Algebraic values of sines and cosines and their arguments

The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these functions are transcendental numbers. Hence, it follows that the values of all angles of the pseudo-Heronian triangle, including the values of all angles of the Pythagoras or Heron triangle, expressed in radians, are transcendental numbers. If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. It should be noted that in this case the argument x = r 2\pi is transcendental and, if expressed in degrees, becomes a rational.

Arithmetic properties of values at polyadic Liouville points of Euler-type series with polyadic Liouville parameter

We study infinite linear independence of polyadic numbers $$𝑓0(𝜆) = ∞Σ 𝑛=0 (𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) = ∞Σ 𝑛=0 (𝜆 + 1)𝑛𝜆𝑛$$, where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 = 𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The result extends the previous author’s result on the polyadic numbers $$𝑓0(1) = ∞Σ 𝑛=0 (𝜆)𝑛, 𝑓1(1) = ∞Σ 𝑛=0 (𝜆 + 1)𝑛$$, The values of generalized hypergeometric series are the subject of numerous studies. If the parameters of the series are rational numbers, then they come either in the class of 𝐸 (if these series are entire functions) or the class of 𝐺 functions (if they have a finite non-zero radius of convergence) or to the class of 𝐹− series ( in the case of zero radius of convergence in the field of complex numbers, however, they converge in the fields of 𝑝− adic numbers). In all these cases, the Siegel-Shidlovsky method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on the Hermite-Pade approximations. In this case, the parameter is a transcendental number. It should be noted that earlier A. I. Galochkin proved the algebraic independence of the values of 𝐸−functions at a point that is a real Liouville number. We also mention the published works of E. Yu. Yudenkova on the values of 𝐹− series in polyadic Liouville points. We especially note that in this paper we consider the values in the polyadic transcendental point of hypergeometric series, the parameter of which is the polyadic transcendental (Liouville) number. Note that earlier A.I. Galochkin proved the algebraic independence of values of 𝐸−functions at points which are real Liouville numbers.We also mention submitted papers (E.Yu.Yudenkova) about the arithmetic properties of values of 𝐹−series at polyadic Liouville numbers. It should be specially mentioned that here we study the values of hypergeometric series with a parameter which is a polyadic Liouville number.

Squaring the Circle and Doubling the Cube in Space-Time

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proven that ⇡ was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space, the impossible becomes possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus, we have added a solution to the impossibility of Doubling the Cube. As a double bonus, we have also Boxed the Sphere! As we will see, one could claim that we have simply bent the rules and moved the problem from one place to another. One of the essential points of this paper, however, is that we can move challenging space problems out from space and into time, and vice versa.

On exremality of affine image of topological product of some varieties

In the theory of Diophantine Approximations one considers a question on approximation of Real Numbers by rational fractions with one and the same denominators. Among intensively studied questions of this theory a special place occupy Metric questions. Here such questions of the theory are considered which take place for almost all real numbers from given interval. For the first time similar questions have been studied by Khintchine for approximation of independent quantities. It had been investigated by him conditions at which for almost all real numbers specified accuracy of approximation is reached. Very important in the technical plan Khintchince’s transference principle allows us to connect a simultaneous approximations of dependent quantities with approximations of linear forms with integral coefficients. In 1932 Mahler K. has entered classification of transcendental numbers into consideration. He showed that almost all transcendental numbers are 𝑆 -numbers. Moreover, Mahler had proved an existence of a constant 𝛾 > 0 such that for almost all 𝜔 $$|𝑃(𝜔)| > ℎ^(−𝑛𝛾)$$, for all polynomials 𝑃 of degree no more 𝑛 and height ℎ > ℎ0(𝜔, 𝑛, 𝛾). Mahler showed that it is possible to take $$𝛾 = 4 + 𝜀$$. In the same work Mahler made an assumption that it is possible to take 𝛾 = 1 + 𝜀 for almost all real numbers. This hypothesis was proved by Sprindzhuk V. G by a method of essential and nonessential domains. Simultaneously, Sprindzhuk V. G. advanced some hypotheses generalising and improving Mahler’s results. Further these investigations of Sprindzuk led to the development of a new direction in the theory of Diophantine Approximations – to the research of extremality of manifolds. In the present article we develop a new approach to these questions and offer a new proof for extremality of algebraic varieties. This method allows to establish extremality of affine image of topological product of some varieties. Considering one particular case, we prove that the extremality for these varieties is possible to deduct from theorems on the convergence exponent of a special integral of Terry’s problem, using E.I. Kovalevskaya’s lemma. Further, we derive from the getting result particular case of the Sprindzuk’s hypothesis on extremality of varieties, induced by monomials of a polynomial in two variables.

On the genesis of BBP formulas

We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for $\pi$. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing $0$ come from. We also explain what is the observed relation between some BBP formulas for $\log 2$ and $\pi$, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transalgebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants.

О коградиентности и контрградиентности линейных дифференциальных уравнений и систем

The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.

Entire functions with undecidable arithmetic properties

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f(z)=∑k=0∞akzk where the coefficients ak∈K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f(α) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f(z) is a transcendental Mahler function, then under generic conditions f(α) is transcendental for all non-zero algebraic numbers with |α|<1. Also, if f(z) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f(n)(α) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f(z), g(z), and h(z) with computable rational coefficients for which no algorithms exist that determine if f(n)∈Q, g(n)(1)∈Q, or ∫01h(z)zndz∈Q for integral n≥0. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9].

On reconstructing subvarieties from their periods

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.

A new transcendental number from the digits of NN

We first give a summary of the history of transcendental numbers, and then we use a nice technique by G. Dresden to find a new transcendental number. In particular, while previous work looked at the last non-zero digit of nn , we consider the digit immediately before its last non-zero digit and show that the infinite decimal built from these digits is transcendental.

Critical base for the unique codings of fat Sierpinski gasket

Given β ∈ (1, 2) the fat Sierpinski gasket is the self-similar set in generated by the iterated function system (IFS) Then for each point there exists a sequence such that , and the infinite sequence (di) is called a coding of P. In general, a point in may have multiple codings since the overlap region has non-empty interior, where Δβ is the convex hull of . In this paper we are interested in the invariant set Then each point in has a unique coding. We show that there is a transcendental number βc ≈ 1.552 63 related to the Thue–Morse sequence, such that has positive Hausdorff dimension if and only if β > βc. Furthermore, for β = βc the set is uncountable but has zero Hausdorff dimension, and for β < βc the set is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of .

Discrete Geometrical Invariants in 3D Space: How Three Random Sequences Can Be Compared in Terms of “Universal” Statistical Parameters

Abstract The statement that any random sequence has a set of deterministic components sounds as absurd and unacceptable. However, based on ideas of Prof. Yu. Babenko, who generalized the Pythagor theorem, it became possible to find a positive answer and state that any random sequence can be expressed quantitatively in terms of the discrete geometrical invariants (DGI). Earlier these DGIs were found for a couple of random sequences on 2D-plane [4,5]. In this paper, the author made a next logical step and obtained the DGI for three random sequences (representing a trajectory of an imaginary particle) in 3D-space. It becomes possible to receive the closed analytical form for the desired DGI of the fourth order in 3D-space and compare a triple of random sequences ({r1k, r2k, r3k}, k=1,2,…N) with each other. This unified and platform identifies (in total) 6 surfaces and 13 reduced and compact parameters having combination from 28 basic moments and their intercorrelations (up to the fourth order, inclusively). This platform reminds the universal form of the partition function proposed by the Gibbs in the statistical physics, when all microscopic parameters describing the trajectories of different micro-particles are transformed to the finite and compact set of thermodynamic variables. Similar idea have been realized earlier with the help of the DGI for a pair of random sequences belonging to 2D-space [4,5]. Based on available data, the author found the 3D images for two famous transcendental numbers as  and E (Euler constant) and their 13 quantitative parameters that differentiate them from each other. Besides, the author applied the DGI approach to analysis of the EQs data. Being a naive user in geophysics, nevertheless it becomes possible to classify available six Earthquakes (EQs) signals and discover their common statistical nature.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Об алгебраических тождествах между фундаментальными матрицами обобщённых гипергеометрических уравнений

В настоящей работе получены примеры алгебраических тождеств между фундаментальнымиматрицами обобщённых гипергеометрических уравнений. В некоторых случаях эти тождествапорождают все алгебраические соотношения между компонентами решенийгипергеометрических уравнений.Обобщённые гипергеометрические функции (см. [1-5]) - это функции вида$${}_l\varphi_{q}(z)={}_l\varphi_{q}(\vec \nu;\vec\lambda;z)={}_{l+1}F_{q}\left(\left.{1,\nu_1,\dots,\nu_l\atop\lambda_1,\dots,\lambda_q}\right|z\right)=\sum_{n=0}^\infty \frac{(\nu_1)_n\dots (\nu_l)_n}{(\lambda_1)_n\dots(\lambda_{q})_n} z^n,$$где $0\leqslant l\leqslant q$, $\; (\nu)_0=1, \; (\nu)_n=\nu(\nu+1)\dots (\nu+n-1)$,$\;\vec\nu=(\nu_1,\dots,\nu_l)\in {\mathbb C}^l$, $\;\vec \lambda\in({\mathbb C}\setminus{\mathbb Z^-})^q$.Функция ${}_l\varphi_{q}(\vec \nu;\vec\lambda;z)$ удовлетворяет(обобщённому) гипергеометрическому дифференциальному уравнению$${L}(\vec \nu;\vec\lambda;z)\;y =(\lambda_1-1)\dots(\lambda_q-1),$$где$${L}(\vec \nu;\vec\lambda;z)\equiv \left( \prod_{j=1}^q(\delta+\lambda_j-1)-z\prod_{k=1}^l(\delta+\nu_k) \right),\label{d1122} \quad \delta=z\frac{d}{dz}.$$В теории трансцендентных чисел одним из основных методов являетсяметод Зигеля-Шидловского (см. [4], [5]), которыйпозволяет доказывать трансцендентность и алгебраическую независимостьзначений целых функций некоторого класса, включающего в себяфункции ${}_l\varphi_{q}(\alpha z^{q-l})$, при условииалгебраической независимости этих функций над ${\mathbb C}(z)$.В статье [6] Ф. Бейкерсом, В. Браунвеллом и Г. Хекманом быливведены важные для установления алгебраической зависимости инезависимости функций понятия коградиентности и контрградиентностидифференциальных уравнений (фактически эти понятия возникли ранеев статье Е. Колчина [7]).Настоящая работа посвящена подробному доказательству и дальнейшемуразвитию результатов о коградиентности и контрградиентности,опубликованных в заметках [8] и [9]. В частности, уточняютсянекоторые результаты статьи [6].

First-Order Orbit Queries

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$ , a square matrix $A\in \mathbb {Q}^{d\times d}$ , and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$ , and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Anx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Arithmetic Properties of Generalized Hypergeometric F-Series

In the paper, using a generalization of the Siegel-Shidlovskii method in the theory of transcendental numbers, we prove the infinite algebraic independence of elements, generated by generalized hypergeometric series, of direct products of the fields of $$\mathbb{K}_v$$-completions of an algebraic number field$$\mathbb{K}$$ of finite degree over the field of rational numbers with respect to a valuation v of the field $$\mathbb{K}$$ extending the p-adic valuation of the field ℚ over all primes p except for finitely many of them.