Theoretical Statistics and Mathematics Unit, ISI Delhi

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

by Kevin G. Hare, Shanta Laishram and Thomas Stoll

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 $n$. We prove and
generalize this conjecture showing that for any polynomial
$p(x)=a_h x^h+a_{h-1} x^{h-1} + \dots + a_0 \in \Z[x]$ 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 $\varepsilon > 0$ we give a bound on the minimal $n$ such that the ratio
$s_q(p(n))/s_q(n) < \varepsilon$. Further, we give lower bounds for the number of
$n < N$ such that $s_q(p(n))/s_q(n) < \varepsilon$.

isid/ms/2011/06 [fulltext]

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