Distribution of the sum-of-digits function of random integers: a survey

Louis H. Y. Chen, Hsien-Kuei Hwang, Vytas Zacharovas

Published 2012-12-30, updated 2014-10-13Version 2

We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.