A lower bound in an approximation problem involving the zeros of the Riemann zeta function

We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called `Hilbert-Polya idea'.Comment: 17 pages, v2 adds two references. No mathematical change

Paley-Wiener spaces with vanishing conditions and Painlev\'e VI transcendents

We modify the classical Paley-Wiener spaces $PW_x$ of entire functions of finite exponential type at most $x>0$, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points $z_1, ..., z_n$. We compute the reproducing kernels and relate their variations with respect to $x$ to a Krein differential system, whose coefficient (which we call the $\mu$-function) and solutions have determinantal expressions. Arguments specific to the case where the "trivial zeros" $z_1, ..., z_n$ are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlev\'e transcendents of the sixth kind as certain quotients of such $\mu$-functions.Comment: 30 page

Deux extensions de Théorèmes de Hamburger (portant sur l’équation fonctionnelle de la fonction zêta)

AbstractWe propose two types of extensions to Hamburger’s theorems on the Dirichlet series with a functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary between the moderate meromorphic functions with the functional equation and the tempered distributions with an extended S-support condition

Sur certains espaces de Hilbert de fonctions entieres, lies a la transformation de Fourier et aux fonctions L de Dirichlet et de Riemann

We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.Comment: 10 pages. In french with an english summar

Des equations de Dirac et de Schrodinger pour la transformation de Fourier

Dyson a associe aux determinants de Fredholm des noyaux de Dirichlet pairs (resp. impairs) une equation de Schrodinger sur un demi-axe et a employe les methodes du scattering inverse de Gel'fand-Levitan et de Marchenko, en tandem, pour etudier l'asymptotique de ces determinants. Nous avons propose suite a notre mise-au-jour de l'operateur conducteur de chercher a realiser la transformation de Fourier elle-meme comme un scattering, et nous obtenons ici dans ce but deux systemes de Dirac sur l'axe reel tout entier et qui sont associes intrinsequement, respectivement, aux transformations en cosinus et en sinus. (Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrodinger equation on the half-axis and has used, in tandem, the Gel'fand-Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms.)Comment: 8 pages, with a summary in English. One or two things adde

Travelling and Standing Waves in a Spatially Forced 2D Convection Experiment

International audienceRayleigh-Bénard convection is studied in a rectangular geometry with a spatial forcing induced in one direction by electric wires. When using fluids of relatively large Prandtl numbers, this forcing allows the existence of a perfect one-dimensional pattern until the onset of bimodal convection. The transition to bidimensional convection is studied for increasing Rayleigh number and reveals the existence of different spatio-temporal regimes depending on the value of the forcing. At the onset of the transition, a stationary pattern is observed for weak forcing, while travelling waves are evidenced for strong forcing. Both behaviours give place to collective oscillations at higher Rayleigh number

Sur les "Espaces de Sonine" associes par de Branges a la transformation de Fourier

Nous avons obtenu des formules explicites representant les fonctions E(z) apparaissant dans la theorie des ``Espaces de Sonine'' associes par de Branges a la transformation de Fourier.Comment: 7 page