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.
"Distribution of the sum-of-digits function of random integers: A survey." Probab. Surveys 11 177 - 236, 2014. https://doi.org/10.1214/12-PS213