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

TL;DR

This survey reviews probabilistic properties of the sum-of-digits function for random integers, introduces new asymptotic approximations, and extends results to general numeration systems using diverse mathematical approaches.

Contribution

It provides a comprehensive overview of existing results and introduces novel asymptotic approximations and methods for analyzing the sum-of-digits function in various numeration systems.

Findings

New asymptotic approximations to total variation distance

Application of four different analytical approaches

Extension of results to general numeration systems

Abstract

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.