Yun Gao, Jining Gao
In this paper, we propose various sufficient conditions to determine if a given real number is an irrational number or a transcendental number and also apply these conditions to some interesting examples, particularly,one of them comes from complex analytic dynamics
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Fast converging irrational series for $ L(2,(\frac d\cdot))$
本文基于 Guillera-Rogers 理论，利用 Kronecker 定理对格和线性组合进行研究，得到了多个 Dirichlet $L$-函数特殊值的几何收敛级数，其适用范围超越了 Guillera-Rogers 的可解情形。
The concept of nearest integer is used to derive theorems and algorithms for the best approximations of an irrational by rational numbers, which are improved with the pigeonhole principle and used to offer an informed presentation of the theory of continued fractions.
In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional continued fraction (with pre--period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences. This multidimensional continued fraction …
For real $ξ$ we consider the irrationality measure function $ψ_ξ(t) = \min_{1\leqslant q \leqslant t, q\in\mathbb{Z}} || qξ||$, where $||\cdot||$ - distance to the nearest integer. We prove that in the case $α\pmβ\notin\mathbb{Z}$ there exist arbitrary large values of $t$ with $$\Bigl | \frac{1}{ψ_α(t)} - \frac{1}{ψ_β(t)} \Bigl | \geqslant \sqrt5\left(1-\sqrt{\frac{\sqrt5-1}{2}}\right)t.$$ The constant on the right-hand side is optimal.
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.
Transcendence criteria for infinite products of algebraic numbers
本文应用 Schmidt 的子空间定理，提出了代数数快速收敛无穷乘积的新超越性判据，并改进了乘积与无穷级数的无理性和超越性判据，推广了 Erdős 的经典定理。
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.
Irrationality of rapidly converging series: a problem of Erdős and Graham
该研究证明了满足双指数增长条件的级数求和结果为无理数，并探讨了其推广形式及最优性。
Let $ξ$ be a real irrational number. We are interested in sequences of linear forms in 1 and $ξ$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $ξ$. We prove that this condition is actually sufficient, even for arbitrary rates of growth …
We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided.
Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadratic irrational parameters
本文研究超几何级数的 $p$-进有界素数密度，将有理参数情形推广至 $_nF_{n-1}$ 并给出精确公式，同时为二次无理参数情形建立下界及条件性精确公式。
We compute the exact irrationality exponents of certain series of rational numbers, first studied in a special case by Hone, by transforming them into suitable continued fractions.
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of rational numbers. This criterion of irrationality is easy to prove and is of great importance. Using it, irrationality of a large class of numbers is proved. We shall apply this method to prove irrationality of algebraic, exponential and trigonometric irrational numbers. Most of the times, explicit constructions are done of the nice sequence of rational numbe…
Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of Fibonacci sequence.
Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational number would be an irrational number. Despite that, it is actually possible for an irrational power of an irrational number to be a rational number. For many generations, this circulated as a non-constructive proof by contradiction in logic for discrete mathematics textbooks and college courses. Since the 80s, a constructive proof circulated orally, such as $\sq…
We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers whose irrationality exponent is not computable.
In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the Diophantine classes and numbers admitting approximations by rational numbers at a given 'speed'. We also show that irrational numbers with finite upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.
Irrationality exponents of logarithms of positive rational numbers
本文通过分离权重行列式论证，证明任意正有理数 r≠1 的对数的无理性指数 μ(log r)=2，并给出完整参数选择与估计，附带几何推广及有限度界。
Following earlier results of Sondow, we propose another criterion of irrationality for Euler's constant $γ$. It involves similar linear combinations of logarithm numbers $L\_{n,m}$. To prove that $γ$ is irrational, it suffices to prove that, for some fixed $m$, the distance of $d\_n L\_{n,m}$ ($d\_n$ is the least common multiple of the $n$ first integers) to the set of integers $\mathbf{Z}$ does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior…
Given quantities $Δ_1,Δ_2,\dots\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<Δ_q$. Depending on the choice of $Δ_q$ and of $x$, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a "metric" point of view, the question is governed by a simple zero--one law: writing $φ$ for Euler's totient function, we either have $\sum_{q=1}^\infty φ(q)Δ_q=\infty$ and then almost all irrational numbers (in the Lebesgue sen…
In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $β{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $ζ(3)$, and the Catalan constant, $G = β(2)$. Interestingly, our series for $\,ζ(3)\,$ converges faster than that used…
Continued fractions and irrationality exponents for modified Engel and Pierce series
本文研究有理数与修正 Engel 或 Pierce 级数组合生成的超越数，利用二阶非线性递推确定其连分数展开，并给出无理性指数下界及可精确计算的无穷族。
Optimal and typical $L^2$ discrepancy of 2-dimensional lattices
该研究针对二维有理及无理格点的$L^2$偏差，通过连分数偏商刻画最优偏差格点并算出已知连分数情形的精确渐近式，同时得到了几乎所有无理格点的偏差渐近性与随机格点的极限分布。
In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.
We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. Consequently we can compute the Lévy constant of any quadratic irrational explicitly in terms of a finite number of its convergents.
From Sturmian and Christoffel words we derive a strictly increasing function $Δ:[0,\infty)\to\mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of $Δ$ distinguishes some irrationality measures of real numbers.
On the irrationality of certain $2$-adic zeta values
本文证明了 Kubota-Leopoldt $2$-adic zeta 函数在特定区间内存在无理数值，通过构造显式有理函数并利用 Volkenborn 积分结合二项式系数操作实现。
Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${μ^{\times}} (ξ)$ is the supremum of the real numbers ${μ^{\times}}$ for which the inequality $$ |b ξ- a|_{p} \leq | a b |^{- {μ^{\times}} / 2} $$ has infinitely many solutions in nonzero integers $a, b$. We show that ${μ^{\times}} (ξ)$ can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of $ξ$. We establish that ${μ^{\times}} ({{ξ_{{\bf t}, p}}}) = 3$, where ${{ξ_{{\bf t}, p}}}$ is the $p$-adic number $1 - p - p^2 + p^…
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.
We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated to the special case $q_i=q^{r_i}, r_i\in\mathbb{N}, q=1/p, p\in\mathbb{N}\setminus\{1\}$. In this construction we make use of little $q$-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of $h^{\pm}(q_1,q_2)$ and give an upper bound for the irrationality measure.
The strong recurrence is equivalent to the Riemann hypothesis. On the other hand, the generalized strong recurrence holds for any irrational number. In this paper, we show the generalized strong recurrence for all non-zero rational numbers. Moreover, we prove that the generalized strong recurrence in the region of absolute convergence holds for any real number.
An effective criterion for periodicity of l-adic continued fractions
针对Ruban的l进连分数，本文给出有理数与二次无理数展开周期性的显式判据，补充有理数情形的定性结果。
Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is motivated by the problem of constructing strict deformation quantizations of symplectic manifolds. We show that this type of approximation exists for any real number and also investigate what happens if the number is rational or a quadratic irrational.
In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{Γ(x)} + \log{Γ(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible exception, but the proof presented there in that work is \emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcend…
It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.
Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group
该研究将超几何工具应用于算术，在含两个奇变量的整数环中定义偶量的幂零“影子”，探究超对称连分数与正交辛模群的概念，为相关性质研究做了初步探索。
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values: $ζ(3),ζ(5),ζ(7),\cdots,ζ(s)$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse …
By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) = LCM(1,...,n), then the fractional part of logS(n) is given by {logS(n)} = d(2n)I(n), for all n sufficiently large, if and only if gamma is a rational number. A corollary is that if {logS(n)} > 1/2^n infinitely often, then gamma is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 0 < n < 2500 and n = 10000) and gamma …
As a corollary of the main result of our recent paper, {\em On the rational approximation of the sum of the reciprocals of the Fermat numbers} published in this same journal, we prove that for each integer $b\geq 2$ the irrationality exponent of $\sum_{n\geqslant 0} s_2(n)/b^n$ is equal to $2$.
On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers
本文研究负连分数下的左 $q$-形变有理数，建立其 $q$-形变 Farey 和计算公式，并据此给出其与有理纽结 Jones 多项式关系的组合证明及同调解释。
We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the Doblin-Lenstra conjecture as well as Jager's subsequent treatment of the sequence of approximation pairs.
Two generalizations of the Minkowski ?(x) function are given. As ?(x) maps quadratic irrationals to rational numbers, it is shown that both generalizations send natural classes of pairs of cubic irrational numbers in the same cubic number field to pairs of rational numbers. It is also shown that these functions satisfy an analog to the fact that ?(x), while continuous and increasing, has derivative zero almost everywhere. Both extend earlier work of Beaver-Garrity on the Farey-Bary map.
For integer $p$, $|p|>1$, and generic rational $x$ and $z$, we establish the irrationality of the series $$\ell_p(x,z)=x\sum_{n=1}^\infty\frac{z^n}{p^n-x}.$$ It is a symmetric ($\ell_p(x,z)=\ell_p(z,x)$) generalization of the $q$-logarithmic function ($x=1$ and $p=1/q$ where $|q|<1$), which in turn generalizes the $q$-harmonic series ($x=z=1$). Our proof makes use of the Hankel determinants built on the Padé approximations to $\ell_p(x,z)$.
We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…
The q-rational numbers and the q-irrational numbers are introduced by S. Morier-Genoud and V. Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of Fibonacci numbers and Pell numbers which are introduced by S. Morier-Genoud and V. Ovsienko. We give an estimation of radiuses of convergence of the…
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