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

Let s q (n) denote the sum of the digits in the g-ary expansion of an integer n. In 1978, Stolarsky showed that $\mathop {lim inf}\limits_{n \to \infty } {{s_2 (n^2 )} \over {s_2 (n)}} = 0$ . He conjectured that, just as for n², this limit infimum should be 0 for higher powers of n. We prove and generalize this conjecture showing that for any polynomial $p(\chi )\,\, = \,\,a_h \chi ^h \, + \,\,a_{h - 1} \chi ^{h - 1} \, + \,...a_o \in \mathbb{Z}[\chi ]$ with h ≥ 2 and a h > 0 any base q, $\mathop {lim inf}\limits_{n \to \infty } {{s_q (n^2 ))} \over {s_q (n)}} = 0$ For any ε > 0 we give a bound on the minimal n such that the ratio s q (p(n))/s q (n) < ε. Further, we give lower bounds for the number of n < N such that s q (p(n))/s q < ε.