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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

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

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 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 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 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

Irrationalité aux entiers impairs positifs d'un q-analogue de la fonction zeta de RiemannDec 11 2007In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s\in\N^* by \zeta_q(s)=\sum_{k\geq 1}q^k\sum_{d|k}d^{s-1}. We give a new lower bound for the dimension of the vector space over \Q spanned, ... More

$L^p$-Spectral theory of locally symmetric spaces with $Q$-rank oneJul 17 2007We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=\Gamma\backslash X$ with finite volume and arithmetic fundamental group $\Gamma$ whose universal covering $X$ is a symmetric space of non-compact ... More

Adiabatic limit of the eta invariant over cofinite quotient of PSL(2,R)May 31 2007We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.

Abstract factorialsMay 29 2007Jul 10 2012A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct ... More

A geometric proof that $e$ is irrational and a new measure of its irrationalityApr 10 2007Oct 06 2010We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e ... More

Irrationality proof of certain Lambert series using little q-Jacobi polynomialsJan 12 2007We 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 ... More

Multiple series connected to Hoffman's conjecture on multiple zeta valuesSep 28 2006Nov 23 2007Recent results of Zlobin and Cresson-Fischler-Rivoal allow one to decompose any suitable $p$-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most $p$; in some cases, only the multiple ... More

Phénomènes de symétrie dans des formes linéaires en polyzêtasSep 27 2006Feb 12 2007We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $\zeta$ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric ... More

Séries hypergéométriques multiples et polyzêtasSep 27 2006We describe a theoretical and effective algorithm which enables us to prove that rather general hypergeometric series and integrals can be decomposed as linear combinations of multiple zeta values, with rational coefficients.

Principal solutions of recurrence relations and irrationality questions in number theoryAug 23 2006Mar 27 2008We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that there exists ... More

On a conjecture of WilfAug 03 2006Jan 26 2007Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to ... More

The Selberg trace formula for Dirac operatorsJul 07 2006We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maass-Laplace operators is also exploited. ... More

Counting and Computing by $e$Jun 26 2006In this paper we count the number of paths and cycles in complete graphs by using the number $e$. Also, we compute the number of derangements in same way. Connection by $e$ yields some nice formulas for the number of derangements, such as $D_n=\lfloor\frac{n!+1}{e}\rfloor$ ... More

Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifoldsJun 09 2006Apr 30 2012For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $\Gamma$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, ... More

Irrationality of $ζ_q(1)$ and $ζ_q(2)$Apr 13 2006In this paper we show how one can obtain simultaneous rational approximants for $\zeta_q(1)$ and $\zeta_q(2)$ with a common denominator by means of Hermite-Pade approximation using multiple little q-Jacobi polynomials and we show that properties of these ... More

Irrationality of some p-adic L-valuesMar 12 2006Jun 20 2006We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using ... More

A Note on Arithmetical Properties of Multiple Zeta ValuesJan 09 2006We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.

Properties of Coefficients of Certain Linear Forms in Generalized PolylogarithmsNov 09 2005We study properties of coefficients of a linear form, originating from a multiple integral. As a corollary, we prove Vasilyev's conjecture, connected with the problem of irrationality of the Riemann zeta function at odd integers.

A family of criteria for irrationality of Euler's constantJul 12 2005Following earlier results of Sondow, we propose another criterion of irrationality for Euler's constant $\gamma$. It involves similar linear combinations of logarithm numbers $L\_{n,m}$. To prove that $\gamma$ is irrational, it suffices to prove that, ... More

An elementary proof of the irrationality of Tschakaloff seriesJun 05 2005We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.

Huber's theorem for hyperbolic orbisurfacesApr 28 2005Jul 30 2007We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the ... More

How can we escape Thomae's relations?Feb 13 2005In 1879, Thomae discussed the relations between two generic hypergeometric $_3F_2$-series with argument 1. It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the ... More

Division Algebras and Non-Commensurable Isospectral ManifoldsJan 05 2005A. Reid showed that if $\Gamma_1$ and $\Gamma_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $\Gamma_1$ and $\Gamma_2$ are commensurable (after ... More

Irrationality of certain p-adic periods for small pAug 16 2004Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations ... More

Tetra and Didi, the cosmic spectral twinsJul 25 2004May 31 2006We introduce a pair of isospectral but non-isometric compact flat 3-manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, ... More

An essay on irrationality measures of pi and other logarithmsApr 29 2004May 27 2004We present a brief survey of the methods used in deducing upper estimates for irrationality measures of the logarithm values. We particularly expose the best known estimates for $\log2$ (due to E. Rukhadze), $\pi$ (due to M. Hata) and $\log3$ (due to ... More