Gérald Tenenbaum

Séries trigonométriques à coefficients arithmétiques

By exploiting the recent analytic tool of friable summation, this work describes a new approach to a class of problems in multiplicative number theory and Fourier series theory originated by H. Davenport. A definite answer to the last original question of Davenport of this type, which was still open, as well as a number of other applications, is given.

Sur une question d'Erdös et Schinzel, II

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

Polynomial values free of large prime factors

For F ∈ Z [ X], let &PSgr; F (x, y) denote the number of positive integers n not exceeding x such that F(n) is free of prime factors > y. Our main purpose is to obtain lower bounds of the form &PSgr; (x, y) >> x for arbitrary F and for y equal to a suitable power of x. Our proofs rest on some results and methods of two articles by the third author concerning localization of divisors of polynomial values. Analogous results for the polynomial values at prime arguments are also obtained.

Fast and strongly localized observation for the Schrödinger equation

We study the exact observability of systems governed by the Schrodinger equation in a rectangle with homogeneous Dirichlet (respectively Neumann) boundary conditions and with Neumann (respectively Dirichlet) boundary observation. Generalizing previous results of Ramdani, Takahashi, Tenenbaum and Tucsnak, we prove that these systems are exactly observable in arbitrarily small time. Moreover, we show that the above results hold even if the observation regions have arbitrarily small measures. More precisely, we prove that in the case of homogeneous Neumann boundary conditions with Dirichlet boundary observation, the exact observability property holds for every observation region whith non empty interior. In the case of homogeneous Dirichlet boundary conditions with Neumann boundary observation, we show that the exact observability property holds if and only if the observation region has an open intersection with an edge of each direction. Moreover, we give explicit estimates for the blow-up rate of the observability constants as the time and (or) the size of the observation region tend to zero. The main ingredients of the proofs are an effective version of a theorem of Beurling and Kahane on non harmonic Fourier series and an estimate for the number of lattice points in the neighbourhood of an ellipse.

Congruences de sommes de chiffres de valeurs polynomiales

Let m, g, q ∈ N with q 2 and (m, q − 1) = 1. For n ∈ N, denote by sq(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients, degree d 1, and such that f(N) ⊂ N, it is shown that there exists C = C(f, m, q) > 0 such that for any g ∈ Z, and all large N, |{0 n N : sq(f(n)) ≡ g (mod m)}| CN min(1,2/d!) . In the special case m = q = 2 and f(n )= n 2 , the value C =1 /20 is admissible. Classification AMS : principale 11B85, secondaires 11N37, 11N69.

Cribler les entiers sans grand facteur premier

Let F(x,y) (resp. F q(x,y)) denote the number of integars at most x (resp. and coprime to q) whose largest prime factor does not exceed y. We give both optimal range of validity and remainder term for the approximation of (resp. Fq(x,y) by ((p(q)/q)lF(x,y). This yields an extension of the range of validity of the smooth approximation of de Bruijn type given by Fouvry and the author for (resp. Fq(x,y).

Sur la non-dérivabilité de fonctions périodiques associées à certaines formules sommatoires

Les fonctions arithmetiques associees aux systemes de representations d’entiers, comme le developpement dans une base donnee, satisfont generalement des relations de recurrence qui facilitent considerablement l’etude de leur valeur moyenne. Considerons par exemple la somme des chiffres en base 2, que nous designons par σ(n).

Sur la répartition des valeurs de la fonction d'Euler

Let Φ(x) denote the number of those integers n with ϕ(n)≤ x, where ϕ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax ≪ R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.

On mean values of random multiplicative functions

Let P denote the set of primes and {f (p)} p∈P be a sequence of independent Bernoulli random variables taking values ±1 with probability 1/2. Extending f by multiplicativity to a random multiplicative function f supported on the set of squarefree integers, we prove that, for any e > 0, the estimate nx f (n) √ x (log log x) 3/2+e holds almost surely—thus qualitatively matching the law of iterated logarithm, valid for independent variables. This improves on corresponding results by Wintner, Erd˝ os and Halasz .

A one-dimensional model with phase transition

AbstractTwo repellent particles are bound to occupy two among thekn+1 adjacent sites 0=x0(n)<x1(n)<...<xkn(n)=1, sayxq(n),xq+1(n). Define the Hamiltonian ℋq(n)=−ln(xq+1(n)−xq(n)) and the partition function We discuss the behaviour of the function