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

TL;DR

This paper proves Stolarsky's conjecture that the ratio of digit sums of polynomial values to original numbers tends to zero, extending the result from squares to all polynomials with positive leading coefficient.

Contribution

It generalizes Stolarsky's conjecture, showing the ratio tends to zero for any polynomial with positive degree, and provides bounds on minimal such n and counts of n below N.

Findings

The ratio s_q(p(n))/s_q(n) approaches 0 as n increases.

Bounds are established for the minimal n with ratio below a threshold.

Lower bounds are given for the count of n less than N with ratio below a threshold.

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 $$ \liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0. $$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + ... + a_0 \in \Z[x]$ with $h\geq 2$ 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 $s_q(p(n))/s_q(n) < \epsilon$.