In the paper discrete limit theorems in the sense of weak convergence of probability measures on the complex plane as well as in the space of analytic functions for the Laplace transform of the Riemann zeta-function are proved.
It is proved that a subspace of a holomorphic Hilbert space is completely determined by their distances to the reproducing kernels. A simple rule is established to localize common zeros of a subspace of the Hardy space of the unit disc. As an illustration we show a series of discs of the complex plan free of zeros of the Riemann -function.
On connaît le lien intime qui existe entre les équations fonctionnelles des fonctions et les formules sommatoires dont le prototype est donné par celle de Poisson. Ce lien fait intervenir la transformation intégrale de Fourier et ses généralisations. Ici, nous réexaminons la signification harmonique (ainsi qu’hilbertienne et distributionnelle) des équations fonctionnelles ayant la forme la plus simple, à savoir, celle s’appliquant pour la fonction dzêta de Riemann et les séries de Dirichlet...
We prove a new type of universality theorem for the Riemann zeta-function and other -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.