Sums of the digits in bases 2 and 3

TL;DR

This paper investigates the relationship between the sum of digits of integers in bases 2 and 3, showing that for large N, most numbers have similar digit sums in these bases within a specific bound.

Contribution

It provides a probabilistic analysis of the digit sums in bases 2 and 3, demonstrating a high concentration of their difference for large integers.

Findings

Over 97% of integers up to N have digit sums in bases 2 and 3 differing by less than approximately 0.146 log n.

The proof relies solely on the marginal distributions of the digit sums in each base.

The result highlights a strong correlation between digit sums in different bases for large numbers.

Abstract

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N 0.970359. The proof only uses the separate (or marginal) distributions of the values of s 2 (n) and s 3 (n).