1,059 results on '"[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]"'
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1002. Tamagawa numbers of diagonal cubic surfaces of higher rank
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Emmanuel Peyre, Yuri Tschinkel, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Peyre, Emmanuel, Arxiv, Import, Tschinkel, Yuri, Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA) more...
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Rank (linear algebra) ,Diagonal ,Picard group ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Rational point ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,14G05 (Primary) 11G35, 14J45 (Secondary) ,Mathematics ,Mathematics - Number Theory ,EAA 020 ,010102 general mathematics ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,Fano variety ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Allgemeines [Mathematik] ,31.00 ,symbols ,Picard horn ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Picard theorem ,Euler product ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We consider diagonal cubic surfaces defined by an equation of the form $$a{{x}^{3}} + b{{y}^{3}} + c{{z}^{3}} + d{{t}^{3}} = 0.$$ Numerically, one can find all rational points of height ≤BforBin the range of up to 105thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previous paper we considered cubic surfaces with Picard groups of rank one with or without Brauer-Manin obstruction to weak approximation. In this paper, we test the conjectures for diagonal cubic surfaces with Picard groups of higher rank. more...
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- 2001
1003. A compactness theorem for the set of uniformly discrete sets and its subsets: lattices, clusters, model sets and Delone sets
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Muraz, Gilbert, Verger-Gaugry, Jean-Louis, and Verger-Gaugry, Jean-Louis
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ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Published
- 2001
1004. Multiple Polylogarithms: An Introduction
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Michel Waldschmidt, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), A. K. Agarwal, Bruce C. Berndt, Christian F. Krattenthaler, Gary L. Mullen, K. Ramachandra and Michel Waldschmidt, Waldschmidt, Michel, and A. K. Agarwal, Bruce C. Berndt, Christian F. Krattenthaler, Gary L. Mullen, K. Ramachandra and Michel Waldschmidt more...
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Physics ,Single variable ,Polylogarithm ,Conjecture ,Mathematics::Commutative Algebra ,Diophantine equation ,Mathematics::Number Theory ,010102 general mathematics ,0102 computer and information sciences ,multiple zeta values ,01 natural sciences ,Unit disk ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,Algebraic relations ,010201 computation theory & mathematics ,Iterated function ,33B30 11G55 11J91 33-02 11M41 ,0101 mathematics ,polylogarithms ,Complex number ,algebraic relations ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Multiple polylogarithms in a single variable are defined by $$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( z \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{{z^{{n_1}}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ , when s1, … , s k are positive integers and z a complex number in the unit disk. For k = 1, this is the classical polylogarithm Li s (z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfy shuffle relations. Multiple polylogarithms in several variables are defined for s i ≥ 1 and |z i | < 1(1 ≤ i ≤ k) by $$L{i_{\left( {{s_1}, \cdots ,{s_k}} \right)}}\left( {{z_1}, \cdots {z_k}} \right) = \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{{z_1^{{n_1}} \cdots z_k^{{n_k}}}} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ , and they satisfy not only shuffle relations, but also stuffle relations. When one specializes the stuffle relations in one variable at z = 1 and the stuffle relations in several variables at z1 = ⋯ = z k = 1, one gets linear or quadratic dependence relations between the Multiple Zeta Values $$\zeta \left( {{s_1}, \cdots ,{s_k}} \right)\sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \geqslant 1} {\frac{1} {{n_1^{{s_1}} \cdots n_k^{{s_k}}}}}$$ which are defined for k, s1, … ,s k positive integers with s1 ≥ 2. The Main Diophantine Conjecture states that one obtains in this way all algebraic relations between these MZV. more...
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- 2000
1005. Séries Gevrey de type arithmétique, I. Th\\éor\èmes de puret\\é et de dualit\\é
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André, Yves, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and arXiv, Import more...
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[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
36 pages, French, published version; Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g. generalized hypergeometric series with rational parameters, enjoy arithmetic counterparts of the Archimedean Gevrey condition; 2) the differential operators which occur in classical treatises on special functions have a rather simple structure: they are either Fuchsian, or have only two singularities, 0 and infinity, one of them regular, the other irregular with a single slope... The main idea of the paper is that the arithmetic property 1) accounts for the global analytic property 2): the existence of an injective arithmetic Gevrey solution at one point determines to a large extent the global behaviour of a differential operator with polynomial coefficients. Proofs use both p-adic and complex analysis, and a detailed arithmetic study of the Laplace transform. more...
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- 2000
1006. Facteurs Q-simples de J_0(N) de grande dimension et de grand rang
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Royer, Emmanuel, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Royer, Emmanuel
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[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
By improvements of J.-P Serre's methods, we prove lower bounds of the largest dimension of simple abelian subvarieties of J0(N), with or without conditions on the algebraic rang (these conditions are : rankX=0 or rankX=dimX)., Par améliorations de méthodes dues à J.-P. Serre, nous donnons des bornes in-férieures de la plus grande dimension des facteurs simples de J 0 (N) soumis à des conditions de rang (ces conditions étant : rang nul ou rang égal à la dimension). Abstract By improvements of J.-P. Serre's methods , we prove lower bounds of the largest dimension of simple abelian subva-rieties of J 0 (N), with or without conditions on the algebraic rank (these conditions are : either rankX = 0 or rankX = dim X). more...
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- 2000
1007. Un demi siècle de transcendance
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Waldschmidt, Michel, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), J.-P. Pier, and Waldschmidt, Michel more...
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transcendence in function fields ,zero lemmas ,$G$-functions ,11Jxx (01A60 11D 11-03) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mahler's method ,rational approximations ,algebraic independence ,method of Siegel-Shidlovskiĭ ,$E$-functions ,Gelfond-Baker's method ,survey ,Gelfond-Schneider's method ,transcendental number theory ,irrationality ,transcendence ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,hypergeometric functions - Abstract
Un rapide survol des progrès de la théorie des nombres transcendants de 1950 à 2000
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- 2000
1008. Effective Polynomial Upper Bounds to Perigees and Numbers of $(3x+d)-$Cycles of a Given Oddlength
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Edward G. Belaga, Thureau, Grégory, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université Louis Pasteur - Strasbourg I, and Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS) more...
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Discrete mathematics ,Combinatorics ,Polynomial ,Algebra and Number Theory ,11K31, 11K38,11K55,11B85 ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mathematics ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
For any positive odd integer d not divisible by 3, define the arithmetical function $T_d(m)$ equal to $mover2$ if m is even, and $3m+d}over2$, otherwise. The $3x+d$ hypothesis, generalizing the well-known $3x+1$ conjecture, asserts that the dynamical system generated by the function $T_d(m)$ has a finite number of cycles and no divergent trajectories. We study here the cyclic structure of this system, and prove in particular an effective polynomial upper bound to the number of cycles with a given number of odd members. more...
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- 2000
1009. On the Eisenstein symbol
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Wildeshaus, Jörg, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Université Sorbonne Paris Cité (USPC)-Centre National de la Recherche Scientifique (CNRS), Bogomolov, Fedor, Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), and Arxiv, Import more...
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Mathematics - Number Theory ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,K-Theory and Homology (math.KT) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-KT] Mathematics [math]/K-Theory and Homology [math.KT] ,11G55 (11G05, 11G07, 11G16, 14D07, 14F20, 14F42, 14F43, 19E08, 19F27) ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Mathematics - K-Theory and Homology ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,FOS: Mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
The main purpose of this paper is the geometric construction, and the analysis of the formalism of elliptic Bloch groups. In the setting of absolute cohomology, we obtain a generalization of Beilinson's Eisenstein symbol to divisors of an elliptic curve, whose support is not necessarily torsion. For motivic cohomology, such a generalization is obtained in low degrees. Our main result shows that the Eisenstein symbol can be defined in all degrees if the elliptic analogue of the Beilinson-Soule vanishing conjecture holds., Comment: 119 pages more...
- Published
- 2000
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1010. 2-extensions with many points
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Semirat, Stephan, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Arxiv, Import more...
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Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,FOS: Mathematics ,Mathematics::Metric Geometry ,Number Theory (math.NT) ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Algebraic Geometry (math.AG) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.
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- 2000
1011. La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs
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Tanguy Rivoal, Structures discrètes et analyse diophantienne (SDAD), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Arxiv, Import, Institut Fourier (IF ), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]) more...
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Pure mathematics ,Mathematics - Number Theory ,11J72 ,Mathematics::Number Theory ,010102 general mathematics ,Mathematics::History and Overview ,010103 numerical & computational mathematics ,General Medicine ,01 natural sciences ,Upper and lower bounds ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Riemann zeta function ,symbols.namesake ,Dimension (vector space) ,Irrational number ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Mathematics ,Vector space - Abstract
We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd integers., Comment: 4 pages (french), accepted for publication (Comptes rendus de l'Academie des sciences de Paris) more...
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- 2000
1012. Conjectures for large transcendence degree
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Waldschmidt, Michel, Waldschmidt, Michel, F. Halter-Koch and R. F. Tichy, Institut de Mathématiques de Jussieu (IMJ), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) more...
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transcendence basis ,transcendence degree ,11J82 11J85 ,algebraic independence of logarithms of algebraic numbers ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; We state two conjectures, which together would yield strong results of algebraic independence related to Schanuel's Conjecture. Partial results on these conjectures are also discussed. more...
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- 2000
1013. Numerical Verification of the Brumer-Stark Conjecture
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Roblot , Xavier-François, Tangedal , Brett, Centre interuniversitaire en calcul mathématique algébrique [Montréal] (CICMA), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM)-Université de Montréal (UdeM), Department of Mathematics, College of Charleston, W. Bosma, CICMA, Concordia University [Montreal, Canada], and Roblot, Xavier-François more...
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[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; We study the Brumer-Stark conjecture computationally in the simplest situation in which it is unproven: an extension K/k with k quadratic, Gal(K/k) isomorphic to ℤ/4ℤ, the class group of K non trivial, K/ℚ non Galois. We verify the conjecture in 379 such cases and study the problem of whether the power of 2 dividing the Brumer element can be replaced by a lower 2-power so that the result remains true. more...
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- 2000
1014. Invitation to higher local fields, Part I, section 18: On ramification theory of monogenic extensions
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Spriano, Luca and Arxiv, Import
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Mathematics - Algebraic Geometry ,Computer Science::Computer Science and Game Theory ,Mathematics - Number Theory ,Mathematics::Complex Variables ,Mathematics::K-Theory and Homology ,Mathematics::Number Theory ,FOS: Mathematics ,12F99, 12J25, 11-99 ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Ramification theory of monogenic extensions of complete discrete valuation fields is presented. Relations to Kato's conductor are discussed., Comment: For introduction and notation, see math.NT/0012131 . Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon3/m3-I-18.abs.html more...
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- 2000
1015. Séries Gevrey de type arithmétique, II. Transcendance sans transcendance
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Yves André, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), arXiv, Import, École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS) more...
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Mathematics (miscellaneous) ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Humanities ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mathematics ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Dans ce second volet, nous etudions les proprietes diophantiennes des valeurs de series Gevrey de type arithmetique d’ordre non nul, en des points algebriques. Nous nous fondons sur le fait, prouve dans le premier volet, que l ’operateur differentiel d ’ordre minimal annulant une telle serie n’a pas de singularites non triviales en dehors de l ’origine et de l ’infini. Nous montrons comment tirer de ce fait des proprietes de transcendance, et retrouver en particulier le theoreme fondamental de la theorie de SiegelShidlovskii sur l’independance algebrique des valeurs de E-fonctions en des points algebriques. Le paradoxe du titre marque le fort constraste entre l’aspect qualitatif de ce nouvel argument, et le caractere eminemment quantitatif des techniques de transcendance traditionnelles (ajustement de parametres auxiliaires...). Dans le cas particulier des valeurs de la fonction exponentielle, nous obtenons une (ou plutot deux) preuve du theoreme de Lindemann-Weierstrass proche de celle “p-adique” de Bezivin-Robba (et qui en est inspiree), ou la transformation de Laplace joue un role-cle. Nous traitons de maniere analogue le cas des series Gevrey d’ordre positif, ou l’on rencontre d’“etranges” formules comme ∑ n≥0 n · n! = −1, valable p-adiquement pour tout p. Enfin, nous discutons brievement le probleme des q-analogues, et prouvons a titre d’illustration deux q-analogues du theoreme de Lindemann-Weierstrass (concernant les valeurs de fonctions theta et de la q-exponentielle respectivement). more...
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- 2000
1016. On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms
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michel waldschmidt, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Arxiv, Import more...
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Mathematics - Number Theory ,11J82 ,General Mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic., Comment: 20 pages. To appear in Publ. Math. Debrecen, 56, 3-4 (2000) See also http://www.math.jussieu.fr/~miw/articles/Debrecen.html more...
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- 2000
1017. De l'euclidianité de $Q(\sqrt{2+\sqrt{2}})$ et $Q(\sqrt{2}\sqrt{2}\sqrt{2}}}) pour la norme
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Cerri, Jean-Paul and Cerri, Jean-Paul
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euclidean minimum ,totally real maximal sub-fields of cyclotomic number fields ,algorithmic ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Nous établissons que les deux corps concernés sont euclidiens pour la norme, de minimum euclidien (et inhomogène) égal à 1/2.
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- 2000
1018. Cyclic Structure of Dynamical Systems Associated with $3x+d$ Extensions of Collatz Problem
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Belaga, Edward G., Mignotte, Maurice, Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS), and Thureau, Grégory more...
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11K38, 11K55,11B85 ,"Collatz problem ,udecidability ,exponential diophantine equations ,udecidability" ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Collatz problem ,iteration of arithmetical functions - Abstract
We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems ${cal D}_d$ generated by iterations of the functions $T_d$ acting, for all $dgeq 1$ relatively prime to 6, on positive integers : $$T_d : {f N} longrightarrow {f N}; qquad T_d(n) = cases{hskip 0.6em elax {n over 2} &, if $n$ is even; {3n+d over 2} &, if $n$ is odd. cr}$$ In the case $d = 1$, the properties of the system ${cal D} = {cal D}_1$ are the subject of the well-known $3x+1$ conjecture. For every one of 6667 systems ${cal D}_d, 1le d le 19999$, we calculate its (complete, as we argue) list of primitive cycles. We unite in a single conceptual framework of primitive memberships, and we experimentally confirm three primitive cycles conjectures of Jeff Lagarias. An in-deep analysis of the diophantine formulae for primitive cycles, together with new rich experimental data, suggest several new conjectures, theoretically studied and experimentally confirmed in the present paper. As a part of this program, we prove a new upper bound to the number of primitive cycles of a given oddlength. more...
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- 2000
1019. Congruence properties of the Ω-function on sumsets
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RIVAT, Joel, Sárközy, András, Stewart, Cameron, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Algebra and Number Theory [Budapest], Eötvös Loránd University (ELTE), University of Waterloo [Waterloo], Rivat, Joel, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU) more...
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ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience
- Published
- 1999
1020. About the Erdos pairs
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Muller, Marie-Paule, Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS), Thureau, Grégory, and Centre National de la Recherche Scientifique (CNRS)-Université Louis Pasteur - Strasbourg I more...
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11B75, 05A16,05D99 ,partitions of integers ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
1) A simpler and more transparent proof is given for earlier results by B. Bollobas and J. Jin about the partitions of an interval of integers into two parts, related to a Ramsey-type problem of Erdos. 2) A new construction allows us to improve some of their results for the problem of Erdos in the more general case of partitions into an arbitrary number of parts. more...
- Published
- 1998
1021. Algorithms for computing isogenies between elliptic curves
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R Lercier, Francois Morain, Lercier, Reynald, D.A. Buell, and J.T. Teitelbaum
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Mathematics::Number Theory ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
The efficient implementation of Schoof's algorithm for computing the cardinality of elliptic curves over finite fields requires the computation of isogenies between elliptic curves. We make a survey of algorithms used for accomplishing this task. When the characteristic of the field is large, P-Weierstrass's functions can be used. When the characteristic of the field is small, we now have three algorithms at our disposal, two due to Couveignes and one to the first author. We treat the same example using these three algorithms and make some comparisons between them. more...
- Published
- 1998
1022. Generalized Meyer Sets and Thue-Morse Quasicrystals with Toric Internal Spaces (invited talk)
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Verger-Gaugry, Jean-Louis, Verger-Gaugry, Jean-Louis, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Michael Baake, Tübingen, Ludwig W. Danzer, Dortmund, Robert V. Moody, Edmonton, and Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS) more...
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Meyer set ,Diffraction Theory ,Cut and project ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Thue-Morse sequence ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Tagungsbericht 16/1998 - APERIODIC ORDER 19.4 - 25.4 1998; International audience; We show that one-dimensional aperiodic sets of points having the Delaunay property can be associated with generalized Meyer sets, for which 1) the internal space is toric, R/lambda Z, with a selection rule based on a congruence mode with respect to the frequencies lambda producing punctuated windows, 2) a scaling exponent in [0,1] can be uniquely defined for each element of the window, related to the scaling properties of the intensity function and the point density measure on canonical 1-dimensional sublattices of period lambda, where a scaling exponent of 1 corresponds to Bragg-peaks, 3) the projection mappings are adapted to the global average lattice and are not orthogonal. The Thue-Morse quasicrystal, arising from the Thue-Morse automaton, is studied modulo lambda Z to show it produces punctuated windows in the one-dimensional torus, for values of lambda selected by congruences. The constructions of Y. Meyer allow arbitrary locally compact Abelian groups as internal spaces but it seems it is the first time reported that the toric compnent is explicitely used linked to the search of scaling and spectral analysis. more...
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- 1998
1023. Galois realizations of families of projective linear groups via cusp forms
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Dieulefait, Luis Victor and Dieulefait, Luis Victor
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Inverse Galois Problem ,Mathematics::Number Theory ,Galois representations ,modular forms ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Using results of Ribet on the images of Galois representations attached to modular forms, we realize linear groups as Galois groups over Q. In particular, combining with results of Brumer we prove that there exist infinitely many exponents r for which PXL_2(p^r) are Galois groups over Q for infinitely many primes p, where PXL means PSL or PGL. (Note: this paper dates from 1998, it is the author's first research work as a PhD student, before the thesis). more...
- Published
- 1998
1024. Reflecting on the $3x+1$ Mystery. Outline of a Scenario
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Belaga, Edward G., Thureau, Grégory, Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Louis Pasteur - Strasbourg I more...
- Subjects
11K31, 11K38,03D35 ,Turing halting problem ,unsolvability ,"iterations of arithmetical functions ,iterations of arithmetical functions ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,unsolvability" ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
"Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a "sure" guess concerning iterations : this is the case of the famous $3x+1$ conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture. " more...
- Published
- 1998
1025. The Hasse principle for the similarities of bilinear forms
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Cortella, Anne, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and Cortella, Anne more...
- Subjects
ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience
- Published
- 1998
1026. Échanges de trois d'intervalles et suites sturmiennes
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Gilles Didier, Institut de mathématiques de Luminy (IML), Centre National de la Recherche Scientifique (CNRS)-Université de la Méditerranée - Aix-Marseille 2, Didier, Gilles, and Université de la Méditerranée - Aix-Marseille 2-Centre National de la Recherche Scientifique (CNRS) more...
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Symbolic dynamics ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,010101 applied mathematics ,Combinatorics ,Number theory ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Mathematics ,Coding (social sciences) - Abstract
International audience
- Published
- 1997
1027. Addendum to the paper of E. Manstavicius and M.N.N Timofeev 'A functional limit theorem related to natural divisors'
- Author
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Tenenbaum, Gérald, Tenenbaum, Gérald, Institut Élie Cartan de Nancy (IECN), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS) more...
- Subjects
limit theorem ,multiplicative functions ,Skorokhod topology ,Distribution of divisors ,AMS Classification: 11K65 (11N64) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
contains corrections with respect to published version.; We provide a simple proof of a result which generalises and makes slightly more precise the main theorem in the paper quoted in the title. This concerns weak convergence, in the Skorokhod space, of distribution functions related to the divisors of a positive integer. more...
- Published
- 1997
1028. Computing M(x)
- Author
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Deléglise, Marc, Rivat, Joel, and Rivat, Joel
- Subjects
ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Published
- 1996
1029. Class number in totally imaginary extensions of totally real function fields
- Author
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Yves Aubry, Laboratoire de Mathématiques Nicolas Oresme ( LMNO ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), and Aubry, Yves more...
- Subjects
Pure mathematics ,Ideal (set theory) ,Tensor product of fields ,010102 general mathematics ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,010103 numerical & computational mathematics ,Extension (predicate logic) ,16. Peace & justice ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,[ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] ,Real-valued function ,Mathematics::Metric Geometry ,Isomorphism ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebra over a field ,Class number ,The Imaginary ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Mathematics - Abstract
International audience; We show that, up to isomorphism, there are only finitely many totally real function fields which have any totally imaginary extension of a given ideal class number.
- Published
- 1996
1030. Estimation du degré d'une isogénie entre courbes elliptiques
- Author
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Pellarin, Federico, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Pellarin, Federico more...
- Subjects
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; Nous estimons le degré d'une isogénie minimale liant deux courbes elliptiques définies sur un corps de nombres k de degré d, en fonction de d et de la hauteur de Weil w de l'une des courbes. On obtient un majorant de la forme cd4 w2 où c est une constante absolue effective. Cela améliore un résultat antérieur de Masser et Wüstholz. more...
- Published
- 1995
1031. The isogeny theorem and the irreducibility theorem for elliptic curves: a survey
- Author
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Pellarin, Federico, Pellarin, Federico, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS) more...
- Subjects
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience
- Published
- 1995
1032. Multidimensional Continued Fractions, Klein-Arnold Sails and Diffraction Theory from Finite and Infinite Icosahedral Clusters
- Author
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Verger-Gaugry, Jean-Louis, Verger-Gaugry, Jean-Louis, and C. Janot, R. Mosseri
- Subjects
ComputingMilieux_MISCELLANEOUS ,[PHYS.COND] Physics [physics]/Condensed Matter [cond-mat] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Published
- 1995
1033. Substitutive normal sets
- Author
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MAUDUIT, Christian, Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Wojciechowska, Anna, and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS) more...
- Subjects
ComputingMilieux_MISCELLANEOUS ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience
- Published
- 1990
1034. Sur une question d'Erdös et Schinzel
- Author
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Gérald Tenenbaum, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Tenenbaum, Gérald more...
- Subjects
010102 general mathematics ,sieve ,010103 numerical & computational mathematics ,Chebyshev problem ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,polynomial values ,0101 mathematics ,11N32 ,divisors in an interval ,Mathematics ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,largest prime factor - Abstract
document différent de la version publiée: de nombreuses erreurs typographiques introduites après l'envoi des épreuves ont été corrigées.; Let F(n) denote a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that y more...
- Published
- 1990
1035. Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations with restricted denominators
- Author
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Rie Natsui, V. Berthé, Hitoshi Nakada, Arithmétique informatique (ARITH), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Department of Mathematics, Japan Women's university, and Berthe, Valerie more...
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Pure mathematics ,Diophantine set ,0102 computer and information sciences ,Rational function ,Diophantine approximation ,Strong law of large numbers ,01 natural sciences ,Theoretical Computer Science ,Law of large numbers ,0101 mathematics ,Laurent formal power series ,Engineering(all) ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,Formal power series ,Diophantine equation ,Applied Mathematics ,010102 general mathematics ,General Engineering ,Metric Diophantine approximation ,16. Peace & justice ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Finite field ,010201 computation theory & mathematics ,Metric (mathematics) ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; We consider metric results for the asymptotic behavior of the number of solutions of Diophantine approximation inequalities with restricted denominators for Laurent formal power series with coefficients in a finite field. We consider, in particular, approximations by rational functions whose denominators are powers of irreducible polynomials, and we study the strong law of large numbers for solutions of the inequalities under consideration more...
- Full Text
- View/download PDF
1036. Symmetric Pascal matrices modulo p
- Author
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Roland Bacher, Robin Chapman, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and arXiv, Import more...
- Subjects
MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,symbols.namesake ,Symmetric polynomial ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Number Theory (math.NT) ,0101 mathematics ,Newton's identities ,Ring of symmetric functions ,Mathematics ,Discrete mathematics ,Power sum symmetric polynomial ,Mathematics - Number Theory ,010102 general mathematics ,Complete homogeneous symmetric polynomial ,Gaussian binomial coefficient ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Computational Theory and Mathematics ,symbols ,11C20, 11T99 ,Elementary symmetric polynomial ,Geometry and Topology ,Combinatorics (math.CO) ,Pascal matrix ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields., Comment: 16 pages, added reference, changes in presentation, correction of an error in a proof more...
- Full Text
- View/download PDF
1037. Substitutions et ensembles normaux
- Author
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Mauduit, Christian, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Université Aix-Marseille II, Gérard Rauzy, Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), and Wojciechowska, Anna more...
- Subjects
Automate Fini ,Oriented graph ,Substitutions ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Given a sequence of integers $u=(u_n)_{n\in\mathbb N}$ generated by a substitution on a finite alphabet, we study the set of real numbers$\alpha$ such that the sequence $(u_n\alpha)_{n\in\mathbb N}$ is equidistributed modulo $1$.This work is related to the study of dynamical systems associated with infinite words generated by substitutions., Si u = (u_n) n∈N est une suite d'entiers engendrée par une substitution sur un alphabet fini, nous étudions l'ensemble des réels α tels que la suite (u_n · α)n∈N soit équirépartie modulo 1. Ce travail est relié à l'étude des systèmes dynamiques associés aux mots infinis engendrés par des substitutions. more...
- Published
- 1989
1038. Automates finis et équirépartition modulo un
- Author
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Mauduit, Christian, Wojciechowska, Anna, Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Université d'Aix-Marseille II, Gérard Rauzy, and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS) more...
- Subjects
Automate fini ,ensemble normal ,finite automata ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Graphe orienté ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Let u = (u_n) n∈N belongs to a sequence of integers recognizable by a finite automata.The purpose of this study is to demonstrate that a sufficient condition for the normal set of "u" to be exactly R ∖Q is that at least one of the vertices that recognizes the sequence "u" is preceded in the automata graph by a vertex having at least two firm paths.This condition can be expressed quantitatively by saying that the sequence "u" must be more "dense" than any exponential seauence., Soit u = (u_n) n∈N est une suite d'entiers reconnaissable par un automate fini. Le but de cette étude est de démontrer qu'une condition suffisante pour que l'ensemble normal de "u" soit exactement R ∖Q est que l'un au moins des sommets qui reconnaît la suite "u" soit précédé dans le graphe de l'automate par un sommet possédant au moins deux chemins fermés.Cette condition peut se traduite quantitativement en disant que la suite "u" doit être plus "dense" que toute suite exponentielle. more...
- Published
- 1984
1039. Répartition modulo 1 des suites automatiques
- Author
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MAUDUIT, Christian, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Université Aix-Marseille II, Gérard Rauzy, Wojciechowska, Anna, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU) more...
- Subjects
Automate Fini ,Finite automata ,Oriented graph ,Graphe orienté ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Let u = (u_n) n∈N belongs to a sequence of integers recognizable by a finite automata.We show that the normal set with respect to u is equal to R∖Q if, and only if, in the oriented graph of the automata, at least one of the vertices which recognize the sequence u is preceded by a vertex from which at least two closed circuits emerge.This condition can be reformulated in qualitatitive terms as follows: the sequence u must be “denser” than any exponential sequence. We deduce the Fourier-Bohr spectrum of any sequence recognizable by a finite automata is included in q., Soit u = (u_n) n∈N est une suite strictement croissante d'entiers reconnaissables par un automate fini.On montre qu'une condition nécessaire et suffisante pour que l'ensemble normal associe a "u" soit exactement R ∖Q est quel'un au moins des sommets qui reconnait la suite "u" soit précédé (au sens large) dans le graphe de l'automate par un sommet possédant au moins deux circuits fermés distincts.Cette condition peut se traduire qualitativement en disant que la suite "u" doit être plus dense que toute suite exponentielle. On en déduit que le spectre de Fourier-Bohr de toute suite reconnaissable par un automate fini est inclus dans q. more...
- Published
- 1986
1040. Sur la répartition des coefficients des formes modulaires de poids demi-entier
- Author
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Darreye, Corentin, Guillaume Ricotta, Florent Jouve, Henri Cohen [Président], Farrell Brumley [Rapporteur], Jie Wu [Rapporteur], Régis de La Bretèche, Emmanuel Royer, Etienne Fouvry, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Bordeaux, and STAR, ABES more...
- Subjects
Half-Integral weight ,Progression arithmétique ,Équirépartition ,Equidistribution ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Signe ,Coefficient de Fourier ,Poids demi-Entier ,Fourier coefficient ,Forme modulaire ,Arithmetic progression ,Modular form ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Sign - Abstract
This thesis deals with some analytic aspects of Fourier coefficients of half-integral weight modular forms. We study in particular two different problems which will be nonetheless connected.On one hand, we are interested in sums of coefficients of half-integral weight cusp forms in arithmetic progressions. Such a problem was studied in a previous paper of Fouvry, Ganguly, Kowalski and Michel for an integral weight cusp form. They showed that, in acertain range of convergence, there is a Gaussian equidistribution of sums of coefficientsin arithmetic progressions of fixed modulus.In this work, we prove an analogous result in the case of a half-integral weight cusp form. We will see that, in a more restricted range of convergence, the sums of coefficients in arithmetic progressions equidistribute with respect to a distribution which is different from the normal distribution obtained by Fouvry, Ganguly, Kowalski and Michel in the integral weight case.On the other hand, we study the signs of Fourier coefficients of a half-integral weight cusp form f and we provide lower bounds for these coefficients. Using techniques from the previous problem and classical results from the theory of half-integral weight modular forms, such as Shimura’s correspondence, Waldspurger’s formula and the recent theory of newforms, we establish a lower bound for the number of normalized coefficients f(n) such that n le x, where n is taken in an arithmetic progression and f(n) > n^{−alpha} for positive alpha., Cette thèse traite de certains aspects analytiques liés aux coefficients de Fourier des formes modulaires de poids demi-entier. On étudie en particulier deux problèmes a priori bien différents mais que l’on reliera.Tout d’abord, on s’intéresse aux sommes des coefficients d’une forme cuspidale de poids demi-entier dans les progressions arithmétiques. Un tel problème fut étudié précédemment dans un article de Fouvry, Ganguly, Kowalski et Michel mais dans le cas d’une forme de poids entier. Les auteurs montrent notamment que, dans un certain régime de convergence, on a une équirépartition gaussienne des sommes des coefficients dans des progressions arithmétiques de module fixé.Dans ce travail, on prouve un résultat analogue lorsque la forme modulaire est de poids demi-entier. On verra que, dans un régime de convergence plus fin, les sommes des coefficients en progression arithmétique s’équirépartissent selon une loi qui est différente de la loi normale obtenue par Fouvry, Ganguly, Kowalski et Michel en poids entier.Dans un deuxième temps, on étudiera les signes des coefficients d’une forme de poids demi-entier f et des possibles minorations en valeur absolue de ces derniers. En utilisant certaines techniques issues du premier problème ainsi que des résultats classiques de la théorie des formes de poids demi-entier, comme la correspondance de Shimura, la formule de Waldspurger ou encore la récente théorie des formes nouvelles, on établie une borne inférieure sur le nombre de coefficients normalisés f(n) tels que n le x, où n est pris dans une progression arithmétique, et f(n) > n^{−alpha} avec alpha > 0. more...
1041. A New and Deterministic Scheme for Characterizing The Organization of Prime Numbers
- Author
-
Hawing, Guilaume M., Patrice OKOUMA, and OKOUMA, Patrice
- Subjects
Number Theory ,Composite Odd Numbers ,Prime Numbers ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
The fundamental theorem of arithmetic states that any composite natural integer can be expressed in one and only one way as a product of prime numbers. This sets the understanding of the organization of prime numbers at the core of number theory. In this work we present a simple, self-consistent and deterministic scheme allowing to investigate further the intrinsic organization of prime numbers. Using this scheme, we establish an algorithm that yields the complete list of prime numbers below any preassigned limit x. Counting the latter yields π(x), the number of prime numbers below x. Based on preliminary tests on computing clusters available, a considerable gain in computational speed and algorithmic simplicity towards producing complete lists of large prime numbers is observed. At the core of the new scheme lays its ability to provide, in a deterministic way, complete lists of consecutive and composite odd numbers below any preassigned limit x. The complete list of prime numbers below x is deduced from the latter. The two key ingredients of the scheme are a set of eleven generic tables, coupled with a three-criteria test applied on the differences between pairs of the consecutive composite odd numbers initially obtained. Since it leads to counting all the elements of a complete list of prime numbers up to x, our deterministic scheme provides a new approach to the long standing problem of " how many prime numbers are there below any preassigned limit x ". The said scheme therefore potentially contributes towards studies aimed at unveiling the organization of prime numbers. We illustrate the latter in a follow-up paper, Paper II [3], where we propose a new perspective on the Riemann hypothesis. more...
1042. On Pythagorean Triples
- Author
-
César Aguilera and Aguilera, César
- Subjects
[MATH] Mathematics [math] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We talk about pythagorean triples and their different representations, we permute these representations to form groups, we classify these groups with a quotient and find that this quotient is related to NSW numbers and primes of the form 2x 2 − y 2 , then we talk about some properties related to all groups, including a pythagorean triples generator, also we talk about linearity, recursion and periodicity, we finally talk about two specific groups and give some identities. more...
1043. Domain of Meromorphy of a class of Bateman-like Totien Zeta-functions
- Author
-
Essouabri, Driss, Velásquez Castañón, Oswaldo, Combinatoire, théorie des nombres (CTN), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), The authors were supported by the MATH-AmSud program: project MZFTTA. O. Velasquez wasalso supported by Peruvian Grant No310-2014-FONDECYT., and Essouabri, Driss more...
- Subjects
meromorphic continuation ,natural boundary ,Bateman-like totien zeta-functions ,11M32, 11M26, 11N37 ,distribution of values of arithmetic functions ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
article en révision; The main result of this paper describe the meromorphic continuationof of a class of Bateman-like totien zeta function associated to a specialclass of arithmetical multiplicative functions, discovering a comb-like form ofthe region of meromorphic continuation, and we verify the expected naturalboundary by clustering logarithmic type singularities. A standard applicationof the Selberg-Delange classical method allows then us to derive in additionthe distribution of values of a class of multivariate multiplicative functions. more...
1044. On Zeta Functions and Tetrahedral Numbers
- Author
-
César Aguilera and Aguilera, César
- Subjects
[MATH] Mathematics [math] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
In this short paper we analyze a possible relation between ζ(n) (for n odd and greater than one) and the Tetrahedral Numbers (number of balls in a triangular pyramid in which each edge contains n balls).[1], also the irrationality of all these (ζ) functions. Introduction. more...
1045. Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$
- Author
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Joseph Muller and MULLER, Joseph
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Mathematics::Algebraic Geometry ,Mathematics - Number Theory ,FOS: Mathematics ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,14G35, 11G18, 14F20, 11E95, 14L05 ,Number Theory (math.NT) ,[MATH.MATH-RT] Mathematics [math]/Representation Theory [math.RT] ,Mathematics::Representation Theory ,Mathematics::Algebraic Topology ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at maximal level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in arXiv:2110.00614 [math.RT]. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and to describe the inertial supports of the irreducible subquotients in the individual cohomology groups. In particular, when $n\geq 5$ no such subquotient is supercuspidal, but when $1\leq n \leq 4$ supercuspidal subquotients do occur. Moreover, we also prove that the cohomology groups need not be admissible in general. Via $p$-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the basic stratum of a Shimura variety with no level at $p$. In the case $n=3$ or $4$, we give a complete description of the cohomology of the basic stratum in terms of automorphic representations. In particular, some automorphic representations occur with multiplicities dependent on $p$., 66 pages. The previous version contained some mistakes. They are now fixed in this new version. The last section and the main theorem have been modified accordingly. Comments are welcome! more...
1046. Hauteur asymptotique des points de Heegner
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Guillaume Ricotta, Thomas Vidick, Département de Mathématiques et de statistique [UdeM- Montréal] (DMS), Université du Québec à Montréal = University of Québec in Montréal (UQAM), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL), and Ricotta, Guillaume more...
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Discretization ,General Mathematics ,Mathematics::Number Theory ,0102 computer and information sciences ,01 natural sciences ,11G50 (Primary) 11M41 (Secondary) ,Quadratic equation ,Principal ideal ,Class number problem ,Courbe elliptique ,FOS: Mathematics ,premier moment ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,Mathematics - Number Theory ,Computer Science::Information Retrieval ,010102 general mathematics ,Mathematical analysis ,formule de Gross-Zagier ,points de Heegner ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Dirichlet character ,11M41, 11G05, 11G40, 11G50, 14G05, 14G10, 14G40 ,fonctions et series L ,Elliptic curve ,hauteurs ,010201 computation theory & mathematics ,Quadratic field ,Heegner number ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
The asymptotic behaviour of the Neron-Tate height of Heegner points on a rational elliptic curve attached to an arithmetically normalized new cusp form f of weight 2, level N and trivial character is studied in this paper. By Gross-Zagier formula, this height is related to the special value at the critical point for the derivative of the Rankin-Selberg convolution of f with a certain weight one theta series attached to some ideal class of some imaginary quadratic field. Asymptotic formula for the first moments asociated to these Dirichlet series are proved and experimental results are carefully discussed., 24 pages, 7 figures, 3 tables, in French, submitted more...
1047. On the X-coordinates of pell equations that are products of two lucas numbers
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Mahadi Ddamulira and Ddamulira, Mahadi
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Physics ,Combinatorics ,Sequence ,Integer ,Lucas number ,Product (mathematics) ,Pell's equation ,Prime (order theory) ,Pell number ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
Let $ \{P_{n}\}_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$, and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers $ d \ge 2$ such that the Pell equations $ x^2-dy^2 = \ell$, where $ \ell\in\{\pm 1, \pm 4\} $, have at least two positive integer solutions $ (x,y) $ and $(x^{\prime}, y^{\prime})$ such that each of $ x$ and $x^{\prime}$ is a product of two Padovan numbers. more...
1048. Questions de localisabilité pour le calcul distribué
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KACHIGAR, Ghazal, Zémor, Gilles, Gavoille, Cyril, Magniez, Frédéric, Perdrix, Simon, Nechita, Ion, Marckert, Jean-François, STAR, ABES, Gilles Zémor, Cyril Gavoille, Jean-François Marckert [Président], Frédéric Magniez [Rapporteur], Simon Perdrix [Rapporteur], and Ion Nechita more...
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[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Théorie des probabilités ,Combinatorics ,Combinatoire ,Probability theory ,Information quantique ,Quantum information theory ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
This thesis is divided in two parts. Its starting point is the concept of resistance to localisation, an important concept in distributed quantum computing.In the first, theoretical part of this thesis, we go over the history of certain concepts and results in quantum information theory and distributed computing, such as the phenomenon of entanglement and the non-signalling condition in the first domain, and the LOCAL model and the colouring problem in the second domain. We then focus on the φ-LOCAL model, whose goal is to study the possibility of quantum distributed algorithms, and which was developedin 2009 by adapting the non-signalling condition to the LOCAL model. We introduce the concepts of global and local consistency in order to emphasise some shortcomings of this model. Finally, we present a more adequate version ofthe φ-LOCAL model.The second part of this thesis contains our major technical results in probability theory. We define the concept of k-localisability which is a probabilistic translation of the φ-LOCAL model. We show that this concept is close to but weaker than the concept of k-dependence which is well-studied in the probabilistic literature. We mention recent results concerning 1-dependent colouring of the path graph and the conclusion they allow us to reach with regards to 1-localisable colouring of the path graph : that it is possible with four or more colours. The rest of this part is dedicated to answering the question of the possibility of 1-localisable colouring of the path graph using three colours which we will show to be impossible. In answering this question we have made use of methods in linear programming and combinatorics. In particular, we prove a theorem on the explicit solution of a linear programming problem having a certain form, and a formula for the Catalan numbers., Cette thèse suit un plan à deux parties. Le point de départ en est la notion de résistance à la localisation, qui est importante en calcul distribué quantique.Dans la première partie, qui est plutôt théorique, nous retraçons l’historique de certaines notions et résultats en information quantique et en calcul distribué, plus précisément le phénomène d’intrication et la condition non-signalling en information quantique et le modèle LOCAL et le problème de coloration en calcul distribué. Ensuite, nous évoquons le modèle φ-LOCAL, développé en 2009 comme adaptation de la condition non-signalling au modèle LOCAL dans le but d’étudier l’existence d’algorithmes distribués quantiques. Finalement, nous soulignons quelques limites du modèle φ-LOCAL à l’aide des notions de consistance globale et de consistance locale, et nous présentons une version plus adéquate de ce modèle.La deuxième partie comporte les principaux résultats techniques obtenus au cours de cette thèse dans le domaine de la théorie des probabilités. Nous introduisons la notion de k-localisabilité qui est une traduction probabiliste du modèle φ-LOCAL. Nous montrons en quoi cette notion est proche, mais plus faible, que la notion de k-dépendance, largement étudiée dans la littérature probabiliste. Nous évoquons des résultats récents autour de la coloration 1-dépendante du chemin qui permettent de conclure au sujet de la coloration 1-localisable du chemin : elle est possible dès qu’il y a plus de quatre couleurs. Dans la suite, nous traitons la question de la possibilité de la coloration 1-localisable du chemin à l’aide de trois couleurs : nous verrons qu’elle n’est pas possible. Pour répondre à cette question, nous avons eu recours à la programmation linéaire et à la combinatoire : en particulier, nous démontrons un théorème qui donne la solution explicite d’un programme linéaire ayant une forme particulière, ainsi qu’une formule pour les nombres de Catalan. more...
1049. Cryptage et décryptage : communiquer en toute sécurité
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Jean-Louis Nicolas, Christophe Delaunay, Nicolas, Jean-Louis, N. Anantharam, J.-M. Bardet, A. de Bouard, A. Gégout-Petit, F. Lagoutière, G. Octavia, Y. Olivier, F. Santambrogio, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), N. Anantharam, J.-M. Bardet, A. de Bouard, A. Gégout-Petit, F. Lagoutière, G. Octavia, Y. Olivier, and F. Santambrogio more...
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courbes elliptiques ,ordinateur quantique ,cryptographie ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
International audience; La sécurisation de nos cartes bleues, ainsi que d'autres procédés de cryptages utilisés couramment, se basent sur l'impossibilité, en pratique, de factoriser de très grands nombres. Ce type de cryptage pourrait cependant être détrôné par d'autres méthodes, sa fiabilité étant sans cesse remise en cause par les progrès de l'informatique. more...
1050. MODULAR CUBIC SEQUENCE PATTERN AND ITS APPLICATION TO THE DIOPHANTINE EQUATION X^3 + Y^3 + Z^3 = K
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Eduardo Acuña, Samuel Flores, Paul Marrero, and Marrero Romero, Paul F.
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Abstract algebra ,sum of three cubes ,Number theory ,Diophantine Equations ,Integer set structure ,[MATH] Mathematics [math] ,Integer sequences ,diophantine ,Mathematical algorithms ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] - Abstract
We propose a new sequence-based algorithm to track and find solutions to the Diophantine equation $X^3+Y^3+Z^3=k$, for a fixed integer $k > 0$.  
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