# GDR STN

Site du GDR Structuration de la théorie des nombres

## Bienvenue sur le site du GDR Structuration de la théorie des nombres

Le GDR Structuration de la théorie des nombres est une unité du CNRS (GDR 2251). Il fédère les recherches en théorie des nombres en France. Il s’investit dans plusieurs activités de caractère national ou international (soutien de rencontres, participation de chercheurs du réseau à des colloques internationaux, mobilité de chercheurs membres ou invités entre pôles du réseau,...), avec une attention particulière aux jeunes chercheurs.

La liste de diffusion du GDR est gdrstn@listes.math.cnrs.fr

Directeur : Emmanuel Royer, Professeur à l’Université Blaise Pascal, emmanuel.royer@math.univ-bpclermont.fr

## Nouveaux articles en théorie des nombres

### [hal-01292805] SUMS OF TWO ${S}$-UNITS VIA FREY-HELLEGOUARCH CURVES

24 mai 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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### [hal-01196990] On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

22 mai 2016

Let $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial $G_n(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_n^−1 )_n≥2$ tends to 1. We prove that the Conjecture of Lehmer is true for $\\theta_n^−1 | n ≥ 2\$ by the direct method of Poincar\'e (...)

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### [cea-01318414] Multiple D3-instantons and mock modular forms I

21 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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### [hal-01319255] Multiple D3-instantons and mock modular forms I

20 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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### [hal-01318564] Central Limit Theorem for probability measures defined by sum-of-digits function in base 2

20 mai 2016

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called µa such that, for any d, µa(d) is the asymptotic density of the set of integers n (...)

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### [hal-01318944] Hecke algebras and affine flag varieties in characteristic p

20 mai 2016

Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant (...)

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### [hal-01314014] Calcul de pi par la méthode originale des carrés inscrits

19 mai 2016

Le nombre pi est utilisé dans des domaines variés des mathématiques, de la physique et d’ingénierie. Le défi que représente la détermination de sa valeur a stimulé la créativité et l'enthousiasme de nombreux chercheurs depuis l'antiquité. De nombreuses méthodes plus ou moins complexes permettent de (...)

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### [hal-01279192] Local behavior of arithmetical functions with applications to automorphic L-functions

19 mai 2016

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang (...)

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### [hal-01280194] Sign changes of Fourier coefficients of modular forms of half integral weight, 2

19 mai 2016

In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.

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### [hal-01278944] The number of coefficients of automorphic L-functions for GL m of same signs

19 mai 2016

Let π be an irreducible unitary cuspidal representation for GL m (A Q), and let L(s, π) be the automorphic L-function attached to π, which has a Dirich-let series expression in the half-plane Re s > 1. When π is self-contragredient, all the coefficients in the Dirichlet series expression are (...)

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### [hal-01278937] Distribution of Hecke eigenvalues of newforms in short intervals

19 mai 2016

We explore nonvanishing of Hecke eigenvalues in short arithmetic progressions and their signs in short intervals by further developing B-free number method and studying moments in short intervals. We improve some results of Alkan & Zaharescu [3] and Lau & Wu [20] on the distribution of (...)

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### [inria-00211875] Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases

19 mai 2016

The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. We prove that for such bases, Kannan's (...)

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### [hal-01278408] On a divisor problem related to the Epstein Zeta-Function, IV

18 mai 2016

In this paper we study several divisor problems related to the Epstein zeta-function. We are able to improve previous results and establish some new results by applying some classical techniques.

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### [hal-01278402] On a divisor problem related to the Epstein Zeta-Function, III

18 mai 2016

In this paper we study the mean square of the error term ∆ * k (Q, x) in a divisor problem related to the Epstein zeta-function. An asymptotic formula has been obtained when k = 2.

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### [hal-01277992] On a divisor problem related to the Epstein Zeta-Function, II

18 mai 2016

Recently by the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and (...)

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### [tel-01316937] Éléments explicites en théorie algébrique des nombres

18 mai 2016

Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en (...)

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### [hal-01316201] Small sumsets in real numbers : a continuous 3k-4 theorem

18 mai 2016

We prove a continuous Freiman's 3k-4 theorem for small sumsets in R by using some ideas from Ruzsa's work on measure of sumsets in R as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when (...)

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18 mai 2016

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### [inria-00187880] LLL: a tool for effective diophantine approximation

18 mai 2016

The purpose of this paper is to survey in a unified setting some of the results in diophantine approximation that the LLL algorithm can make effective in an efficient way. We mostly study the problems of finding good rational approximations to vectors of real and p-adic numbers, and of finding (...)

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18 mai 2016

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### [inria-00000904] Euclidean minima of totally real fields : Algorithmic determination.

18 mai 2016

This article deals with the determination of the Euclidean minimum $M(K)$ of a totally real number field $K$ of degree $n\geq 2$, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree $2$ to $8$ and (...)

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18 mai 2016

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### [hal-00091199] Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2

18 mai 2016

Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée

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### [hal-00136940] Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1

18 mai 2016

Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M(K) is the Euclidean minimum of K, we have . This phenomenon has consequences on the decidability of the Euclidean nature (...)

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### [hal-00136941] Euclidean minima of totally real number fields, algorithmic determination

18 mai 2016

Cet article traite de la détermination explicite du minimum euclidien (pour la norme) des corps de nombres totalement réels, voire de la partie supérieure de leur spectre euclidien. L'efficacité de l'algorithme mis en oeuvre permet de travailler jusqu'au degré (...)

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### [hal-00107309] LLL on the average

18 mai 2016

Despite their popularity, lattice reduction algorithms remain mysterious in many ways. It has been widely reported that they behave much more nicely than what was expected from the worst-case proved bounds, both in terms of the running time and the output quality. In this article, we (...)

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### [hal-01282675] HIGHER INTEGRAL MOMENTS OF AUTOMORPHIC L-FUNCTIONS IN SHORT INTERVALS

17 mai 2016

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

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### [hal-01282682] Distribution of values of symmetric power L-functions at the edge of the critical strip

17 mai 2016

We study some problems on the distribution of values of symmetric power L-functions at s = 1 in both aspects of level and weight: bounds of these values, extreme values, Montgomery-Vaughan's conjecture and distribution functions. Our results generalise and/or improve related results of (...)

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### [tel-01282669] Valeurs centrales et valeurs au bord de la bande critique de fonctions L automorphe

17 mai 2016

Cette thèse, constitué en trois parties, est consacrée à l’étudie des valeurs spéciales de fonctions L automorphes. La première partie contient un survol rapide de la théorie des formes modulaires et des fonctions L de puissance symétrique associées qui est nécessaire dans la suite. Dans la seconde (...)

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### [hal-01316252] A note on generalized radix representations and dynamical systems

16 mai 2016

Akiyama et al. proved an asymptotic formula for the distribution of CNS poly-nomials with fixed constant term. The objective of the present paper is to improve that result by providing an error term too.

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### [hal-01316251] Construction of Normal Numbers via Generalized Prime Power Sequences

16 mai 2016

We construct normal numbers in base q by concatenating q-adic expansions of prime powers $\left\lfloor\alpha p^\theta\right\rfloor$ with $\alpha>0$ and $\theta>1$.

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### [hal-01316249] Construction of μ -normal sequences

16 mai 2016

In the present paper we extend Champernowne's construction of normal numbers to provide sequences which are generic for a given invariant probability measure, which need not be the maximal one. We present a construction together with estimates and examples for normal numbers with respect to (...)

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### [hal-00875091] On the distribution functions of two oscillating sequences

16 mai 2016

We investigate the set of all distribution functions of two special sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterise the set of all distribution functions $G(x_n)$ for $(x_n)_n \geq 1 = (\\cos (\alpha n)^n\)_n \geq 1$ and arbitrary (...)

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### [hal-01314745] Sign changes of the coefficients of automorphic L-functions

13 mai 2016

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients of modular forms, which includes the great contributions of other authors. We are attempting to elucidate interesting viewpoints, ideas and methods which, even unspecified, may (...)

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12 mai 2016

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### [hal-01315080] Multidimensional Bell inequalities and quantum cryptography. Codes, cryptology, and information security.

12 mai 2016

The laws of quantum physics allow the design of cryptographic protocols for which the security is based on physical principles. The main cryptographic quantum protocols are key distribution schemes, in which two parties generate a shared random secret string. The privacy of the key can be (...)

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### [hal-01315075] A qutrit quantum key distribution protocol using Bell inequalities with larger violation capabilities.

12 mai 2016

The Ekert quantum key distribution protocol [1] uses pairs of entangled qubits and performs checks based on a Bell inequality to detect eavesdropping. The 3DEB protocol [2] uses instead pairs of entangled qutrits to achieve better noise resistance than the Ekert protocol. It performs checks (...)

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12 mai 2016

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### [hal-01315047] Estimates of ψ,θ for large values of x without the Riemann hypothesis

12 mai 2016

The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual (...)

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### [hal-01314997] On the splitting of the exact sequence, relating the wild and tame kernels

12 mai 2016

Let k be a number field. For an odd prime p and an integer i>1, the i-th \'etale wild kernel is contained in the second cohomology group of o'_k with coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa theory, we give conditions for this inclusion to split. In (...)

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12 mai 2016

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10 mai 2016

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### [hal-01313346] Shifted convolution of cusp-forms with θ-series

10 mai 2016

In this paper we apply simple approachs to improve a recent result due to Luo, concerning a shifted convolution sum involving the Fourier coefficients of cusp forms with those of theta series. In order to explore what should be the best possible bound, a mean square result of this shifted (...)

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### [hal-01307789] There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31

10 mai 2016

We proved that $\liminf\limits_n \rightarrow +\infty(p_n+1-p_n)=2$ where $p_n$ is the $n-th$ prime number. We showed the conditions on which an integer $10X+1$, $10X+3$, $10X+7$ or $10X+9$ can be a prime number. We studied the conditions required to get some twin primes and proved the twin (...)

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10 mai 2016

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### [hal-01281342] Sur la répartition du noyau d'un entier

10 mai 2016

We investigate the asymptotic behaviour of the number $N(x,y)$ of those integers $n\leqslant x$ with squarefree kernel $k(n)\leqslant y$.

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### [hal-01281305] Mesures quadratiques de la proximité des diviseurs

10 mai 2016

Diverses mesures naturelles de la proximité des diviseurs d'un entier sont introduites et plusieurs conjectures sont résolues.

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### [hal-01313347] Fourier coefficients of symmetric power L-functions

10 mai 2016

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group SL 2 (Z). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish some bounds for (...)

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### [hal-01280840] On irregularities of distribution of binary sequences relative to arithmetic progressions, I. (General results)

10 mai 2016

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic (...)

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### [hal-01280845] On a phenomenon of Turán concerning the summands of partitions

10 mai 2016

Turán [12] proved that for almost all pairs of partitions of an integer, the proportion of common parts is very high, that is greater than 1/ 2 − ε with ε > 0 arbitrarily small. In this paper we prove that this surprising phenomenon persists when we look only at the summands in a fixed (...)

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### [hal-01280829] Various approaches for the study of the complexity of some families of pseudorandom subsets

10 mai 2016

Studying randomness in different structures is important from the development of applications and theory. Dartyge, Mosaki and Sárközy (among others) have studied measures of randomness for families of subsets of integers. In this article, we improve results on the complexity of some families (...)

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### [hal-01303279] Primes in [kn, (k + 1)n] , for 1⩽. k ⩽. 10

9 mai 2016

In 1919 S. Ramanujan proved [Ram] the Bertrand postulate elegantly with approximations based on the Stirling formula. P. Erdös proved also [Erd] the same result in 1931 by a carefull study of the binomial coeficient (2n n). The Bertrand postulate was first proved by P. Tchebichef in 1850. (...)

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### [hal-01312603] Diophantine approximation with improvement of the simultaneous control of the error and of the denominator

9 mai 2016

In this work we proof the following theorem which is, in addition to some other lemmas, our main result: \noindent \textbftheorem. Let$\ X=\left\ \left( x_1\text, % t_1\right) \text, \left( x_2\text, t_2\right) \text, ..., \left( x_n\text, t_n\right) \right\$ be a finite part of (...)

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### [hal-01311299] Lower Bounds for the Height in Galois Extensions

9 mai 2016

We prove essentially sharp lower bounds for the height of an algebraic number in a Galois extension

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### [hal-01081309] Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques

8 mai 2016

We provide, in a wide range of the parameters, an estimate for the mean-value over sifted integers of certain arithmetic functions with Dirichlet series analytically close to $1/\zeta_\mathbbK(s)$, where $\mathbbK$ is a number field and $\zeta_\mathbbK$ its Dedekind zeta (...)

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7 mai 2016

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### [hal-01281714] Sur l'inégalité de Turán–Kubilius friable

7 mai 2016

Nous établissons une version uniforme, relativement à tous les paramètres, de l’inégalité de Turán-Kubilius friable (i.e.relative aux entiers sans grand facteur premier), comparant la variance empirique d’une fonction arithmétique additive à support friable à celle de son modèle probabiliste. Nous (...)

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### [hal-01281724] An Ingham-Müntz type theorem and simultaneous observation problems

7 mai 2016

We establish a theorem combining the estimates of Ingham and Müntz – Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz – Szász part. A very special case of this theorem allows us to prove the simultaneous (...)

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7 mai 2016

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### [hal-01281732] Fonctions multiplicatives, sommes d'exponentielles, et loi des grands nombres

7 mai 2016

Nous donnons des conditions suffisantes effectives quasi-optimales pour que la majoration de Halberstam–Richert fournisse effectivement l'ordre de grandeur de la valeur moyenne d'une fonction multiplicative positive ou nulle.

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5 mai 2016

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### [hal-01067949] On the generic local Langlands correspondence for GSpin groups

5 mai 2016

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to (...)

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### [hal-01296795] Application de réciprocité de Shafarevich et groupe de Nottingham

4 mai 2016

Soient K le corps local View the MathML source et Kab sa clôture abélienne. On note View the MathML source le groupe des Fq-automorphismes continus σ de Kab tel que σ(K)=K et tel que σ opère sur le corps résiduel View the MathML source comme une puissance entière de l'automorphisme de Frobénius. On (...)

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### [hal-01311391] Sums of seventh powers of polynomials over a finite field with 8 elements

4 mai 2016

Let F be a finite field with 8 elements. We study representations of polynomials over F as sums and strict sums of seventh powers.

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### [hal-01311367] Sums of seventh powers in the ring of polynomials over the finite field with four elements

4 mai 2016

We study representations of polynomials P in F4[T] as sums P = X17 + . . . + Xs7.

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### [hal-01311352] Waring's problem for polynomials in two variables

4 mai 2016

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers. We also give bounds for the number of terms $s$ and the (...)

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### [hal-01311343] Sums of $(p^r+1)$-th powers in the polynomial ring $\Bbb F_{p^m}[T]$

4 mai 2016

Let p be an odd prime number and let F be a finite field with p(m) elements. We study representations and strict representations of polynomials M epsilon F[T] by sums of (p(r) + 1)-th powers. A representation M = M-1(k) + ... + M-s(k) of M epsilon F[T] as a sum of k-th powers of polynomials is (...)

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### [hal-01311330] On Murata densities

4 mai 2016

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci. 56(7) (1980) 351-353; On some fundamental relations among certain asymptotic densities, Math. Rep. (...)

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### [hal-01311322] Sums of seventh powers in the polynomial ring $\mathbb{F}_{2^{m}}[T]$

4 mai 2016

Let F be a finite field with even characteristic and q >= 16 elements. We study representations of polynomials P is an element of F [T] as sums P = X-1(7) + ... + X-s(7).

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4 mai 2016

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### [hal-01272342] Sur les puissances des polynômes sur un corps fini

4 mai 2016

Let F a finite field, Q a given element of F[T] with positive degree and k a positive integer. The goal of this work is to study the Q-automaticity of the set of kth powers of elements of F[T] and to calculate the number of polynomials X ∈ F[T] of degree N such that the sum of digits of Xk in (...)

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### [hal-01309134] Are monochromatic Pythagorean triples avoidable?

4 mai 2016

A Pythagorean triple is a triple of positive integers a, b, c ∈ N-0 satisfying a^2 + b^2 = c^2. Is it true that, for any finite coloring of N-0, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X^2 +Y^2 = Z^2 regular? This problem has been open (...)

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### [hal-01311138] Weak Néron models for cubic polynomial maps over a non-Archimedean field

4 mai 2016

The aim of this note is to give an effective criterion to verify whether a cubic polynomial over a non-Archimedean field has a weak N\'eron model or not.

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### [tel-01310561] Gauss's theorem on sums of 3 squares, sheaves, and Gauss composition

4 mai 2016

Gauss's theorem on sums of 3 squares relates the number of primitive integer points on the sphere of radius the square root of n with the class number of some quadratic imaginary order. In 2011, Edixhoven sketched a different proof of Gauss's theorem by using an approach from arithmetic (...)

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### [hal-00995770] Inverse binomial series and values of Arakawa-Kaneko zeta functions

3 mai 2016

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two classes of zeta functions of Arakawa-Kaneko-type by means of certain inverse binomial series involving harmonic sums which appeared (...)

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### [hal-00604393] Le produit harmonique des suites

3 mai 2016

Au moyen d'une transformation binomiale involutive dans l'espace des suites à valeurs complexes, on définit un nouveau produit dénommé " harmonique " en raison de ses remarquables propriétés à l'égard des sommes harmoniques. La transformation d'Euler des séries permet de déduire de ces (...)

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### [hal-00495767] A new class of identities involving Cauchy numbers, harmonic numbers and zeta values

3 mai 2016

Improving an old idea of Hermite, we associate to each natural number $k$ a modified zeta function of order $k$. The evaluation of the values of these functions $F_k$ at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta (...)

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### [hal-00499548] Relations entre les sommes d'Euler de poids fixé et les valeurs de zêta

3 mai 2016

Dans cet article, on montre comment déduire d'une remarquable formule sommatoire d'Ohno une relation non-triviale entre les sommes d'Euler de poids fixé $k$ et $\zeta(k)$. In this article, we show how to derive from a deep sum formula of Ohno a non-trivial relation between Euler sums of fixed (...)

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### [hal-01202725] Produit harmonique, sommation de Ramanujan et fonctions zêta d'Arakawa-Kaneko

3 mai 2016

Ce mémoire s'articule autour des trois thèmes principaux auxquels j'ai consacré mes recherches au cours de ces 15 dernières années, thèmes qui sont assez étroitement reliés entre-eux comme je m'attacherai à le montrer. Il s'agit du produit harmonique, du procédé de sommation de Ramanujan et de la (...)

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### [inria-00395839] Noether's forms for the study of non-composite rational functions and their spectrum

3 mai 2016

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the (...)

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### [hal-01226725] Sur le radical kummérien des Zl-extensions

22 avril 2016

Sur la base d'un travail antérieur, nous donnons une description nouvelle du radical iniial attaché au compositum des Zl-extensions d'un corps de nombres en termes de limites projectives pour la norme dans la Zl-extension cyclotomique de ce corps. Le résultat précis que nous obtenons contient, (...)

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### [hal-01280172] On Fourier coefficients of modular forms of half integral weight at squarefree integers

21 avril 2016

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s>\tfrac12$. This exhibits a high fluctuation of the coefficients at squarefree (...)

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### [hal-00131641] Points de hauteur bornee sur les varietes de drapeaux en caracteristique finie

20 avril 2016

The aim of this paper is to apply the work of Morris on Eisenstein series over global function fields to the study of the asymptotic behavior of the points of bounded height on a generalized flag variety defined as the quotient of a semi-simple algebraic group by a reduced parabolic subgroup (...)

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### [hal-01304309] Ramification in Iwasawa Theory and Splitting Conjectures

19 avril 2016

We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalized class groups and generalized Jacobians. We (...)

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### [hal-01263097] A NOTE ON MULTIPLE ZETA VALUES IN TATE ALGEBRAS

19 avril 2016

In this note, we shall discuss a generalization of Thakur's multiple zeta values and allied objects, in the framework of function fields of positive characteristic and more precisely, of periods in Tate algebras.

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### [hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

19 avril 2016

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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### [hal-01277623] On shifted Mascheroni Series and hyperharmonic numbers

17 avril 2016

In this article, we study the nature of the forward shifted series σ r = n>r |bn| n−r where r is a positive integer and b n are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ (−k) of zeta at the negative integers and Euler's constant γ. These expressions may (...)

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### [hal-01302600] On the average distribution of divisors of friable numbers

16 avril 2016

A number is said to be y-friable if it has no prime factor greater than y. In this paper, we prove a central limit theorem on average for the distribution of divisors of y-friable numbers less than x, for all (x, y) satisfying 2 ≤ y ≤ e (log x)/(log log x) 1+ε. This was previously known under the (...)

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### [hal-01302604] Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

16 avril 2016

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As (...)

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### [hal-01283042] Series Representation of Power Function

16 avril 2016

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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### [hal-01283042] Series Representation of Power Function

16 avril 2016

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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### [hal-00771342] Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

15 avril 2016

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this (...)

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15 avril 2016

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### [hal-00911140] Improvements on the accelerated integer GCD algorithm

29 mars 2016

The present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's ''Accelerated integer GCD algorithm'' is (...)

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### [hal-00689464] Some remarks on barycentric-sum problems over cyclic groups

29 mars 2016

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements g_1, ... (...)

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### [hal-01292805] SUMS OF TWO S-UNITS VIA FREY-HELLEGOUARCH CURVES

25 mars 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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### [hal-01293564] Vectorial Drinfeld modular forms over Tate algebras

25 mars 2016

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[θ])). Among several results that we prove here, we determine the complete (...)

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### [hal-01293573] A note on certain representations in characteristic p and associated functions

25 mars 2016

We discuss certain representations of GL 2 Fq[T] in equal characteristic and associated vectorial modular forms

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### [hal-01293687] On a torsor of paths of an elliptic curve minus a point

25 mars 2016

We are studying some aspects of the action of the Galois groups on torsors of paths on an elliptic curve minus a point. We construct objects whose behaviour is similar to the classical polylogarithms on the projective line minus three (...)

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### [hal-01293667] Non-abelian unipotent periods and monodromy of iterated integrals. (2003) 2(1),

25 mars 2016

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on P1ℚ(μn)∖0,μn,∞. Let n be a prime number. We assume that for any m≥1 the numbers Lim+1(ξkn) for 1≤k≤(n−1)/2 are linearly independent over ℚ in ℂ/(2π\ri)m+1ℚ. Let S=k1,⋯,kq be a subset of 1,…,p−1 such that if k∈S, (...)

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25 mars 2016

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### [hal-01293611] Cosimplicial objects in algebraic geometry ,

25 mars 2016

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincaré groupoid) over X × X. (...)

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### [hal-01293604] On functional equations of p-adic polylogarithms. Bull. Soc. Math. France , 119 (1991)

25 mars 2016

ABSTRACT . — The n-t h order polylogarithm Ln(z) i s defined by th e series E^Ll zk ''t^ n on tn e Q? 611 un it disc. This functio n has multivalued analytic prolon- gation t o C \ 0,1 . The same series ^^ z k f^ defines an analytic p-adic functio n on th e open uni t disc in Cp (a completion o (...)

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25 mars 2016

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24 mars 2016

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### [hal-01292754] Milnor K-theory and the graded representation ring

24 mars 2016

Let F be a field, let G = Gal(¯ F /F) be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k * (F) to the graded ring gr R(G, k) associated to Grothendieck's (...)

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### [hal-01293244] Central extentions and coverings .Publications Universitat Autonoma de Barcelona. vol. 29 (1985), 145-153.

24 mars 2016

The theory of central extensions has a lot of analogy with the theory of covering spaces. In this paper we show that the category of central extensions of a perfect group and a certain category of covering spaces are equivalent. Then the facts about central extensions will follow from the (...)

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24 mars 2016

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24 mars 2016

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### [hal-01292727] A functoriality principle for blocks of p-adic linear groups

24 mars 2016

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. (...)

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### [hal-00371233] On the multiplicative order of $a^n$ modulo $n$

24 mars 2016

Let $n$ be a positive integer and $\alpha_n$ be the arithmetic function which assigns the multiplicative order of $a^n$ modulo $n$ to every integer $a$ coprime to $n$ and vanishes elsewhere. Similarly, let $\beta_n$ assign the projective multiplicative order of $a^n$ modulo $n$ to every integer (...)

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### [hal-01291833] The Fourier expansion of $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$

23 mars 2016

We compute the Fourier coefficients of the weight one modular form $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form (...)

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### [hal-00371228] On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

23 mars 2016

Let $X$ be a finite sequence of length $m\geqslant 1$ in $\mathbbZ/n\mathbbZ$. The derived sequence $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length $1$ is reached, (...)

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### [hal-00823402] On special values of spinor L-functions of Siegel cusp eigenforms of genus 3

22 mars 2016

We compute the special values for the spinor L-function L(s,F12) in the critical strip s=12,...,19, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of (...)

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### [hal-01160765] Microsolutions of differential operators and values of arithmetic Gevrey series

22 mars 2016

We continue our investigation of E-operators, in particular their connection with G-operators; these differential operators are fundamental in understanding the dio-phantine properties of Siegel's E and G-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian (...)

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