On Riemann sums

Abstract

The study of almost sure convergence of Riemann sums is a fascinating question which has connections with various problems from Number Theory, among them the Riemann hypothesis through its link with Farey sequences. Moreover, it has been known since the fundamental paper of Rudin, that the convergence almost everywhere of Riemann sums, along a given subsequence of positive integers, definitively relies on the arithmetical properties of the subsequence. The arithmetical characterization of that property is an open and certainly hard question. The study of Riemann sums has for years been an object of constant interest from analysts, ergodicians, and number theorists. It even seems, that its power of attraction has grown even more during this last decade. This is the reason of the present survey. Our motivation in writing it, was to propose a text to the interested reader, giving a direct access to the main results of that theory, as well as an easy understanding, as far as possible each time in each case, of the various methods elaborated by the authors of these results.