Stolarsky’s conjecture and the sum of digits of polynomial values
By
Kevin G. Hare, Shanta Laishram and Thomas Stoll
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→∞ = 0. He conjectured that, 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(x) = a_hx^h + a_h-1x^h-1 + … + a₀ [x] with h ≥ 2 and a_h > 0 and any base q,

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(n) < ε.