It is well known that value at a non-zero algebraic number of each of the functions $e^{x}, \ln x, \sin x, \cos x, \tan x, \csc x, \sec x, \cot x, \sinh x,$ $ \cosh x,$ $ \tanh x,$ and $\coth x$ is transcendental number (see Theorem 9.11 of \cite{N}). In the work, we show that for any one of the above mentioned functions, $f(x)$, and for a polynomial $g(x)$ with rational coefficients the zero, if any, of the equation $f(x)=g(x)$ is a transcendental number. We also show that if $f(x)$ and $g(x)$ are polynomials with rational coefficients, then a zero of the equation $e^{f(x)}=g(x)$ is a transcendental number. Finally we show that the existence of an abelian group whose non-zero elements are transcendental numbers.
We generalize Lindemann-Weierstrass theorem and Gelfond -Schneider-Baker Theorem. We find new transcendental numbers in this work. There are several methods to find transcendental numbers in the work. Recently transcendental numbers are applicable for cryptography (\cite{G}, \cite{K}, \cite{V}). Since we are able to make many tables of random numbers, the new transcendental numbers will be applicable for encryption and decryption in this work (\cite{V}, \cite{Z}).
To decide upon the arithmetic nature of some numbers may be a non-trivial problem. Some cases are well know, for example exp(1) and W(1), where W is the Lambert function, are transcendental numbers. The Tsallis q-exponential, e_q (z), and the Lambert-Tsallis W_q (z) function, where q is a real parameter, are, respectively, generalizations of the exponential and Lambert functions. In the present work we use the Gelfond-Schneider theorem in order to show the arithmetic conditions on q and z such that W_q (z) and exp_q (z) are transcendental.
In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…
We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the Ax logarithmique theorem and the Ax-Lindemann theorem which we shall prove. We also prove the whole Ax-Schanuel conjecture for the unipotent part. All these bi-algebraic and transcendental results generalize their counterparts for mixed Shimura varieties. In the end we briefly discuss about…
Given a subset $S=\{s_0, s_1\}$ of the complex plane with two points and an infinite subset ${\mathscr S}$ of $S\times {\mathbb N}$, where ${\mathbb N}=\{0,1,2,\dots\}$ is the set of nonnegative integers, we ask for a lower bound for the order of growth of a transcendental entire function $f$ such that $f^{(n)}(s)\in{\mathbb Z}$ for all $(s,n)\in{\mathscr S}$. We first take ${\mathscr S}=\{s_0,s_1\}\times 2{\mathbb N}$, where $2{\mathbb N}=\{0,2,4,\dots\}$ is the set of nonnegative even integers. We prove that an entire function $f$ of sufficiently small exponential type such that $f^{(2n)}(s_…
The need to evaluate Logarithmic integrals is ubiquitous in essentially all quantitative areas including mathematical sciences, physical sciences. Some recent developments in Physics namely Feynman diagrams deals with the evaluation of complicated integrals involving logarithmic functions. This work deals with a systematic review of logarithmic integrals starting from Malmsten integrals to classical collection of Integrals, Series and Products by I. S. Gradshteyn and I. M. Ryzhik [1] to recent times. The evaluation of these types of integrals involves higher transcendental functions (i.e., Hur…
Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non vanishing integral global section of an hermitian line bundle over $X$ is either zero or it cannot be too small with respect to the $\sup$ norm of the section itself. We study inequalities similar to Liouville's for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture…
We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy's extremal numbers, Bugeaud-Laurent Sturmian continued fractions, and more generally the class of Sturmian type numbers. We compute, for each real number $ξ$ of this set, several exponents of Diophantine approximation to the pair $(ξ,ξ^2)$, together with $ω_2^*(ξ)$ and $\widehatω_2^*(ξ)$, the so-called ordinary and uniform exponent of approximation to $ξ$ by algebraic numbers of degree $\leq 2$. As an application, we get new information on the set of values taken by $\widehatω_2^…
Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$ rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies $f^{(mn+r_j)}(s_j)\in{\mathbb Z}$ for $0\le j\le m-1$ and all sufficiently large $n$, then $f$ is a polynomial. Under suitable assumptions on $s_0,s_1,\dots,s_{m-1}$ and $r_0,\dots,r_{m-1}$, we introduce interpolation polynomials $Λ_{nj}$, ($n\ge 0$, $0\le j\le m-1$) satisfying $$ Λ_{nj}^{(mk+r_\ell)}(s_\ell)=δ_{j\ell}δ_{nk}, \quad\hbox{for}\quad n, k\ge 0 \quad\hbox{an…
We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some ``trivial zeros'' exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those sp…
In previous papers we investigated basic properties of the determinant $G_{K}(s)$ of the Riemann operator: ${\mathcal R}$ acting on $\bigoplus_{n>1} K_{n}(A)_{\mathbb{C}}$, where $A$ is the integer ring of an algebraic number field $K$. The function $G_{K}(s)$ is defined as the regularized determinant \[ G_{K}(s) = {\rm det} ((s I-\mathcal{R}) | \bigoplus_{n>1} K_{n}(A)_{\mathbb{C}} ) \] with $\mathcal{R} | K_{n}(A)_{\mathbb{C}} = \frac{1-n}{2}$. We showed that $G_{K}(s)^{-1}$ is essentially the so called gamma factors of Dedekind zeta function of $K$. In this paper we study the transcendency …
Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $φ:M\to X({\Bbb C})$ be a holomorphic map. For every positive real number $T$, let $A_U(T)$ be the cardinality of the set of $z\in U$ such that $φ(z)\in X(K)$ and $h_L(φ(z))\leq T$. After a revisitation of the proof of the sub exponential bound for $A_U(T)$, obtained by Bombieri and Pila , we show that there are intervals of $T$'s as big as we want for which $A_U(T)$ is upper bounded by a polynom…
We will apply Nevanlinna Theory to prove several Ax-Schanuel type Theorems for functional transcendence when the exponential map is replaced by other meromorphic functions. We also show that analytic dependence will imply algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax-Schanuel Theorem will be discussed.
We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a connected pure Shimura variety with a fixed quasiprojective embedding. We show that there is an explicitly computable function $B$ of two natural number arguments so that for any field extension $K$ of the complex numbers and Hodge generic irreducible proper subvariety $X \subsetneq S_K$, the set of nonconstant points in the intersection of $X$ with the unio…
