Stolarsky's Conjecture and Sum of Binary Digits of $n$ and $n^2$

Abstract: Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $\displaystyle{ \liminf_{n\to\infty}} \frac{s_2(n^2)}{s_2(n)} = 0$. He conjectured that, as for $n^2$, this limit infimum should be $0$ for higher powers of $n$. In this talk,I will give a survey of the some of the results on the distribution of $s_q(n)$ and give an idea of the proof of Stolarsky's conjecture. Also I will discuss some results on the structure of $n$ for which $s_2(n)=s_2(n^2)$. This is a joint work with Kevin Hare at Waterloo and Thomas Stoll at Marseille.