total 162took 0.11s

Fields of dimension one algebraic over a global or local field need not be of type $(C_{1})$May 16 2019Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ with the following properties: (i) $E$ has dimension dim$(E) \le 1$, i.e. the Brauer group Br$(E ^{\prime ... More

Roots of $L$-functions of characters over function fields, generic linear independence and biasesMar 13 2019We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ... More

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

Rational lines on cubic hypersurfacesSep 21 2018We show that any smooth projective cubic hypersurface of dimension at least $29$ over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to ... 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

Arithmetic properties of series of reciprocals of algebraic integersJul 12 2018Dec 18 2018We obtain results bounding the degree of the series $\sum_{n=1}^{\infty} 1/\alpha_n$, where $\{\alpha_n\}$ is a sequence of algebraic integers satisfying certain algebraic conditions and growth conditions. Our results extend results of Erd\H{o}s, Han\v{c}l ... More

A Simple Criterion for Irrationality of Some Real NumbersJun 20 2018Jul 29 2018In this paper, we calculate the limit of the average of the decimals of some real numbers. For example, we show that the limit of the average of the decimals of simply normal numbers is 9/2. We also prove that if a real number $r$ cannot be represented ... 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

On the real projections of zeros of almost periodic functionsMay 05 2018This paper deals with the set of the real projections of the zeros of an arbitrary almost periodic function defined in a vertical strip $U$. It provides practical results in order to determine whether a real number belongs to the closure of such a set. ... 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

Arithmetical properties of real numbers related to beta-expansionsAug 10 2017The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with ... 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.

The Product $e π$ Is IrrationalJun 19 2017Jun 01 2018This note shows that the product $e \pi$ of the natural base $e$ and the circle number $\pi$ is an irrational number.

On Padé approximations and global relations of some Euler-type seriesMay 11 2017We shall consider some special generalizations of Euler's factorial series. First we construct Pad\'e approximations of the second kind for these series. Then these approximations are applied to study global relations of certain p-adic values of the series. ... More

Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau and Shen)Apr 26 2017Mar 27 2019About 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

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

On a result of Fel'dman on linear forms in the values of some E-functionsApr 06 2017We shall consider a result of Fel'dman, where a sharp Baker-type lower bound is obtained for linear forms in the values of some E-functions. Fel'dman's proof is based on an explicit construction of Pad\'e approximations of the first kind for these functions. ... More

On Mahler's transcendence measure for $e$Apr 05 2017May 02 2018We present a completely explicit transcendence measure for $e$. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of ... 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

Hyperbolic triangles without embedded eigenvaluesFeb 19 2014Nov 04 2017We 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 2013May 27 2017A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,\Theta_1,...,\Theta_m\in\mathbb{C}^*$ over the ring $\mathbb{Z}_{\mathbb{I}}$ of an imaginary quadratic field $\mathbb{I}$. This work deals with the simultaneous ... 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

Irrationality of the Zeta ConstantsDec 13 2012Jun 22 2018A 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