Stolarsky's conjecture and the sum of digits of polynomial values

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 $$\underset{n\to \infty }{\mathrm{liminf}}\frac{s_2(n^2)}{s_2(n)}=0.$$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h{x}^h+a_{h-1}{x}^{h-1}+...+a_0\in \mathbb{Z}[x]$ with $h\ge 2$ and $a_h>0$ and any base $q$, \[ \liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.\] For any $\epsilon >0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n)<\epsilon$. Further, we give lower bounds for the number of $n<N$ such that $s_q(p(n))/s_q(n)<\epsilon$.

13 pages