Latest in 11j72

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Irrationality and transcendence of continued fractions with algebraic integersFeb 12 2019We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their ... More
The equivalence principle for almost periodic functionsJan 22 2019Given two arbitrary almost periodic functions, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition ... More
Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cuspsDec 13 2018In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We ... More
A Linear independence result for $p$-adic $L$-valuesSep 20 2018The celebrated theorem of Ball--Rivoal gives a lower bound for the dimension of the $\mathbb{Q}$-vector space $\mathbb{Q}+\mathbb{Q}\zeta(3)+...+\mathbb{Q}\zeta(s)$ spanned by the odd positive zeta values between $3$ and $s$. In particular, they have ... More
On a conjecture of LivingstonAug 29 2018In an attempt to resolve a folklore conjecture of Erd\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the $\overline{\mathbb{Q}}$ ... More
Zero is a resonance of every Schottky surfaceAug 28 2018For certain spectral parameters we find explicit eigenfunctions of transfer operators for Schottky surfaces. Comparing the dimension of the eigenspace for the spectral parameter zero with the multiplicity of topological zeros of the Selberg zeta function, ... More
On linear relations for Dirichlet series formed by recursive sequences of second orderMay 08 2018Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by \[\zeta_F(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad \zeta_F^*(s) \,:=\,\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{F_n^s}\,,\quad ... More
Counting cusp forms by analytic conductorMay 02 2018The universal family is the set of cuspidal automorphic representations of bounded analytic conductor on ${\rm GL}_n$ over a number field. We prove an asymptotic for the universal family, under a spherical assumption at the archimedean places when $n\geqslant ... More
Arithmetic of Catalan's constant and its relativesApr 26 2018We prove that at least one of the six numbers $\beta(2i)$ for $i=1,\dots,6$ is irrational. Here $\beta(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes Dirichlet's beta function, so that $\beta(2)$ is Catalan's constant.
Some hypergeometric integrals for linear forms in zeta valuesApr 11 2018We prove integral representations of the approximation forms in zeta values constructed in arXiv:1801.09895 and arXiv:1803.08905.
Many odd zeta values are irrationalMar 23 2018Apr 09 2018Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number ... More
Infinitely many odd zeta values are irrational. By elementary meansFeb 26 2018In this small note, we provide an elementary proof of the fact that infinitely many odd zeta values are irrational. For the first time, this celebrated theorem been proven by Rivoal and Ball--Rivoal. The original proof uses highly non-elementary methods ... More
Hypergeometry inspired by irrationality questionsFeb 24 2018Aug 05 2018We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement is a new partial ... More
Metrical irrationality results related to values of the Riemann $ζ$-functionFeb 12 2018We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost ... More
One of the Odd Zeta Values from $ζ(5)$ to $ζ(25)$ Is Irrational. By Elementary MeansJan 30 2018Mar 29 2018Available proofs of result of the type 'at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however ... More
Diophantine triples in linear recurrence sequences of Pisot typeNov 10 2017The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness ... More
Remainder Padé approximants for the Hurwitz zeta functionSep 15 2017Following our earlier research, we use the method introduced by the author in \cite{prevost1996} named Remainder Pad\'e Approximant in \cite{rivoalprevost}, to construct approximations of the Hurwitz zeta function. We prove that these approximations are ... More
What is $\ldots\ $ a multiple orthogonal polynomial?Jul 29 2017This is an extended version of our note in the Notices of the American Mathematical Society 63 (2016), no. 9, in which we explain what multiple orthogonal polynomials are and where they appear in various applications.
Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau and Shen)Apr 26 2017About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the ... More
Euler's factorial series and global relationsMar 07 2017Using Pad\'e approximations to the series $E(z)=\sum_{k=0}^\infty k!(-z)^k$, we address arithmetic and analytical questions related to its values in both $p$-adic and Archimedean valuations.
Determinants of Laplacians on Hilbert modular surfacesJan 23 2017We study regularized determinants of Laplacians acting on the space of Hilbert-Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in [4,5].
Prime geodesic theorem of Gallagher typeJan 09 2017We reduce the exponent in the error term of the prime geodesic theorem for compact Riemann surfaces from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure.
On Koyama's refinement of the prime geodesic theoremJan 06 2017We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are ... More
Linear combinations of prime powers in sums of terms of binary recurrence sequencesDec 18 2016Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain the finiteness ... More
Toric surfaces over an arbitrary fieldOct 20 2016Sep 13 2018We study toric varieties over an arbitrary field with an emphasis on toric surfaces in the Merkurjev-Panin motivic category of "K-motives". We explore the decomposition of certain toric varieties as K-motives into products of central simple algebras, ... More
Intrinsic sound of anti-de Sitter manifoldsSep 20 2016As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in ... More
The sup-norm problem for GL(2) over number fieldsMay 30 2016Oct 02 2018We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature ... More
The sup-norm problem for GL(2) over number fieldsMay 30 2016We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature ... More
On linear independence measures of the values of Mahler functionsApr 06 2016In this paper, we estimate the linear independence measures for the values of a class Mahler functions of degree one and two. For the purpose, we study the determinants of suitable Hermite-Pad\'{e} approximation polynomials. Based on the non-vanishing ... More
Riemann-Roch isometries in the non-compact orbifold settingApr 01 2016Sep 22 2016We generalize work of Deligne and Gillet-Soul\'e on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $\Gamma\backslash\mathbb{H}$, for $\Gamma\subset PSL_{2}(\mathbb{R})$ a fuchsian group of ... More
Diophantine triples with values in $k$-generalized Fibonacci sequencesFeb 26 2016We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of $\{F_n^{(k)}: ... More
On the irrationality of generalized $q$-logarithmJan 11 2016Jan 24 2016For 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 ... More
A fast modulo primes algorithm for searching perfect cuboids and its implementationJan 04 2016A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their non-existence ... More
Hankel Determinants of Zeta ValuesOct 07 2015Dec 17 2015We study the asymptotics of Hankel determinants constructed using the values $\zeta(an+b)$ of the Riemann zeta function at positive integers in an arithmetic progression. Our principal result is a Diophantine application of the asymptotics.
Only finitely many Tribonacci Diophantine triples existAug 31 2015Jan 20 2016Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here ... More
A determinantal approach to irrationalityJul 21 2015Feb 02 2016It is a classical fact that the irrationality of a number $\xi\in\mathbb R$ follows from the existence of a sequence $p_n/q_n$ with integral $p_n$ and $q_n$ such that $q_n\xi-p_n\ne0$ for all $n$ and $q_n\xi-p_n\to0$ as $n\to\infty$. In this note we give ... More
On simultaneous approximation of the values of certain Mahler functionsMay 05 2015Jun 27 2016In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Pad\'{e} approximations of the functions under consideration, then apply the functional equations to get an infinite ... More
Applications of differential algebra to algebraic independence of arithmetic functionsApr 10 2015Jan 16 2017We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature. ... More
Applications of differential algebra to algebraic independence of arithmetic functionsApr 10 2015Sep 02 2015We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature. ... More
Notes on noncommutative geometryMar 17 2015The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises is attached. ... More
Hankel determinants, Padé approximations, and irrationality exponentsMar 10 2015Sep 01 2015The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained ... More
Rational Angled Hyperbolic PolygonsDec 11 2014We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle ... More
Rationality problem for quasi-monomial actionsNov 11 2014We give a short survey of the rationality problem for quasi-monomial actions which includes Noether's problem and the rationality problem for algebraic tori, and report some results on rationality problem in three recent papers Hoshi, Kang and Kitayama ... More
Rationality and powerSep 12 2014We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed algebraic number. ... More
Local average in hyperbolic lattice point countingAug 25 2014Oct 13 2016The hyperbolic lattice point problem asks to estimate the size of the orbit $\Gamma z$ inside a hyperbolic disk of radius $\cosh^{-1}(X/2)$ for $\Gamma$ a discrete subgroup of $\hbox{PSL}_2(R)$. Selberg proved the estimate $O(X^{2/3})$ for the error term ... More
A few remarks on values of Hurwitz Zeta function at natural and rational argumentsMay 24 2014Dec 08 2014We exploit some properties of the Hurwitz zeta function $\zeta (n,x)$ in order to study sums of the form $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}1/(jk+l)^{n}$ and $\frac{1}{\pi ^{n}}\sum_{j=-\infty}^{\infty}(-1)^{j}/(jk+l)^{n}$ for $% 2\leq n,k\in ... More
Towards the (ir)rationality of values of Dirichlet seriesApr 10 2014We show that if $F(s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F(k)$ is a rational number for all large enough positive integers $k$, then the denominators of those rational numbers are unbounded. In particular, ... More
Hyperbolic triangles without embedded eigenvaluesFeb 19 2014We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study ... More
On simultaneous diophantine approximations to $ζ(2)$ and $ζ(3)$Jan 21 2014May 17 2014We present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, ... More
Bounds for eigenforms on arithmetic hyperbolic 3-manifoldsJan 21 2014Apr 13 2015On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By ... More
Some transcendence results from a harmless irrationality theoremOct 28 2013Feb 07 2014The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values ... More
An explicit Baker type lower bound of exponential valuesSep 24 2013Let $\mathbb{I}$ denote an imaginary quadratic field or the field $\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring of intergers. We shall prove an explicit Baker type lower bound for $\mathbb{Z}_{\mathbb{I}}$-linear form of the ... More
On Baker type lower bounds for linear formsSep 23 2013We wish to give an axiomatic approach to (explicit) Baker type lower bounds for linear forms, over the ring $\mathbb{Z}_{\mathbb{I}}$ of an imaginary quadratic field $\mathbb{I}$, of given numbers $1,\Theta_1,...,\Theta_m\in\mathbb{C}^*$. In this work ... More
Beukers-like proofs of irrationality for $ζ{(2)}$ and $ζ{(3)}$Aug 12 2013May 13 2016In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit square integrals proposed originally by Beukers, I introduce some modifications which certainly ... More
Dynamical invariants for group automorphismsJun 25 2013We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given and the problem ... More
Euler's constant: Euler's work and modern developmentsMar 07 2013Oct 25 2013This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes ... More
Some conjectures in elementary number theoryFeb 21 2013We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
Poisson-Newton formulas and Dirichlet seriesJan 28 2013Jan 29 2013We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton ... More
The Schanuel Subset Conjecture implies Gelfond's Power Tower ConjectureDec 31 2012Nov 26 2013As an alternative to the famous Schanuel's Conjecture (SC), we introduce the Schanuel Subset Conjecture (SSC): Given $\alpha_1,...,\alpha_n\in \mathbb{C}$ linearly independent over $\mathbb{Q}$, if $\{\alpha_1,...,\alpha_n, e^{\alpha_1},...,e^{\alpha_n}\}$ ... More
Irrationality of the Zeta ConstantsDec 13 2012Sep 16 2016A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the zeta constants ... More
Asymptotics of analytic torsion for hyperbolic three--manifoldsDec 13 2012Feb 13 2018We prove that for certain sequences of hyperbolic three--manifolds with cusps which converge to hyperbolic three--space in a weak ("Benjamini-Schramm") sense and certain coefficient systems the regularized analytic torsion approximates the $L^2$-torsion ... More
Asymptotics of analytic torsion for hyperbolic three--manifoldsDec 13 2012Jul 09 2013We prove that for certain sequences of hyperbolic three--manifolds with cusps which converge to hyperbolic three--space in a weak ("Benjamini-Schramm") sense and certain coefficient systems the regularized analytic torsion approximates the $L^2$-torsion ... More
Poincaré series for non-Riemannian locally symmetric spacesSep 18 2012Dec 11 2013The discrete spectrum of the Laplacian has been extensively studied on reductive symmetric spaces and on Riemannian locally symmetric spaces. Here we examine it for the first time in the general setting of non-Riemannian, reductive, locally symmetric ... More
Double Dirichlet series and quantum unique ergodicity of weight 1/2 Eisenstein seriesSep 10 2012Oct 14 2014The problem of quantum unique ergodicity (QUE) of weight 1/2 Eisenstein series for {\Gamma}_0(4) leads to the study of certain double Dirichlet series involving GL2 automorphic forms and Dirichlet characters. We study the analytic properties of this family ... More
A new construction of the real numbers by alternating seriesAug 07 2012Oct 30 2013We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there exist infinitely ... More
Explicit bounds on canonical Green functions of modular curvesJul 25 2012We prove explicit bounds on canonical Green functions of Riemann surfaces obtained as compactifications of quotients of the upper half-plane by Fuchsian groups.
On an incomplete argument of Erdos on the irrationality of Lambert seriesJun 02 2012We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.
An approximate spectral representation and explicit bounds on Green functions of Fuchsian groupsMay 29 2012Jul 19 2012We study the Green function gr_\Gamma\ for the Laplace operator on the quotient of the hyperbolic plane by a cofinite Fuchsian group \Gamma. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of ... More
On new rational approximants to ζ(3)Apr 30 2012New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A comparison of ... More
Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constantFeb 14 2012Apr 05 2012We rebut Kowalenko's claims in 2010 that he proved the irrationality of Euler's constant, and that his rational series for it is new.
Nesterenko's linear independence criterion for vectorsFeb 10 2012Oct 01 2013In this paper we deduce a lower bound for the rank of a family of $p$ vectors in $\R^k$ (considered as a vector space over the rationals) from the existence of a sequence of linear forms on $\R^p$, with integer coefficients, which are small at $k$ points. ... More
An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic SurfacesOct 10 2011Nov 17 2017We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric ... More
An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic SurfacesOct 10 2011Mar 08 2012We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric ... More
Rational points on singular intersections of quadricsAug 09 2011May 14 2012Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the Gaussians.
Note On the Irrationality of the L-Function Constants L(s, X)May 10 2011Oct 12 2012A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.
The irrationality of a number theoretical seriesMay 07 2011Denote by $\sigma_k(n)$ the sum of the $k$-th powers of the divisors of $n$, and let $S_k=\sum_{n\geq 1}\frac{\sigma_k(n)}{n!}$. We prove that Schinzel's conjecture H implies that $S_k$ is irrational, and give an unconditional proof for the case $k=3$. ... More
The irrationality of some number theoretical seriesMay 07 2011We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.
Linear independence measures for values of certain q-seriesFeb 10 2011We prove, in a quantitative form, linear independence results for values of a certain class of q-series, which generalize classical q-hypergeometric series. These results refine our recent estimates.
Stripes on rectangular tilingsJan 16 2011We consider a class of cut-and-project sets $\Lambda = \Lambda_F \times \zahl$ in the plane. Let $L=\Lambda+w\real$, $w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a dense subset of ... More
On the resonances of convex co-compact subgroups of arithmetic groupsNov 29 2010Let $\Lambda$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=\Lambda \backslash\H$ has infinitely many resonances in an effective ... More
Symmetries of the transfer operator for $Γ_0(N)$ and a character deformation of the Selberg zeta function for $Γ_0(4)$Nov 19 2010Jan 19 2011The transfer operator for $\Gamma_0(N)$ and trivial character $\chi_0$ possesses a finite group of symmetries generated by permutation matrices $P$ with $P^2=id$. Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm ... More
Dynamics of tuples of matrices in Jordan formMar 27 2010Dec 20 2011A tuple (T_1,...,T_k) of (n x n) matrices over R is called hypercyclic if for some x in R^n the set {T^{m_1} T^{m_2}...T^{m_k} x : m_1,m_2,...,m_k in N} is dense in R^n. We prove that the minimum number of (n x n) matrices in Jordan form over R which ... More
Dissolving cusp forms: Higher order Fermi's Golden RulesMar 14 2010Sep 27 2010For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to become resonances. A sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermi's Golden Rule. We prove formulas ... More
On the criteria for linear independence of Nesterenko, Fischler and ZudilinDec 24 2009In 1985, Yu. V. Nesterenko produced a criterion for linear independence, which is a variant of Siegel's. While Siegel uses upper bounds on full systems of forms, Nesterenko uses upper and lower bounds on sufficiently dense sequences of individual forms. ... More
Recurrent proofs of the irrationality of certain trigonometric valuesNov 10 2009We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions ... More
Irrationality proofs à la HermiteNov 10 2009Aug 16 2010As rewards of reading two great papers of Hermite from 1873, we trace the historical origin of the integral Niven used in his well-known proof of the irrationality of $\pi$, uncover a rarely acknowledged simple proof by Hermite of the irrationality of ... More
Irrationality exponent and rational approximations with prescribed growthOct 23 2009Let $\xi$ be a real irrational number. We are interested in sequences of linear forms in 1 and $\xi$, 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 ... More
Report on some recent advances in Diophantine approximationAug 27 2009A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well ... More
Research note on a well posed integral used in Apery's proof for the irrationality of Z(3)Jul 06 2009In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function
Rational numbers with purely periodic $β$-expansionJul 01 2009Jan 22 2010We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field ... More
An Asymptotic relation for Hadjicostas FormulaJun 27 2009We derive an asymptotic formula which in some cases generalize Hadjicostas formula
Arithmetic theory of q-difference equations (G_q-functions and q-difference modules of type G, global q-Gevrey series)Feb 24 2009Jan 13 2010In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, \S 13]. In the second part of the ... More
Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifoldsJan 26 2009We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\Gamma,\Sigma}^{\rm o}(\lambda)$ associated to the spinor bundle $\Sigma$ on odd dimensional convex co-compact hyperbolic manifolds $X:=\Gamma\backslash\hh^{2n+1}$. ... More
On the non-quadraticity of values of the q-exponential function and related q-seriesDec 15 2008We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\lambda)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-\lambda)}, \qquad |q|>1, \quad \lambda\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff ... More
On a linear form for Catalan's constantOct 10 2008It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8\phi_7$-series and a balanced $_4\phi_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on ... More
Irrationality proof of a $q$-extension of $ζ(2)$ using little $q$-Jacobi polynomialsSep 15 2008Sep 18 2008We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater ... More
Some sufficient conditions of a given series with rational terms converging to an irrational number or a transcdental numberJul 09 2008Jul 18 2008In 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 ... More
A Geometric Proof to Cantor's Theorem and an Irrationality Measure for Some Cantor's SeriesApr 29 2008Dec 25 2010Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series.
Mertens' theorem for toral automorphismsJan 14 2008May 27 2010A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbits ... More
On the irrationality of Ramanujan's mock theta functions and other q-series at an infinite number of pointsDec 24 2007We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm 1/2,\pm 1/3,\pm ... More