On a second conjecture of Stolarsky: the sum of digits of polynomial values

Let q, r ≥ 2 be integers, and denote by s _q the sum-of-digits function in base q. In 1978, K.B. Stolarsky conjectured that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \frac{s_2(n^r)}{s_2(n)} \leq r.$$ In this paper we prove this conjecture. We show that for polynomials ${P_1(X), P_2(X) \in \mathbb{Z}X]}$ of degrees r ₁, r ₂ ≥ 1 and integers q ₁, q ₂ ≥ 2, we have $$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))} = \frac{r_1 (q_1 - 1) {\rm log}q_2}{r_2(q_2 - 1) {\rm log} q_1}.$$ We also present a variant of the problem to polynomial values of prime numbers.