Sur une conjecture de Stolarsky et la somme des chiffres des valeurs polynomiales

International audienceLet s_q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim inf n→∞ 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 ∈ Z[x] with h ≥ 2 and a h > 0 and any base q, lim inf n→∞ s_q (p(n))/ 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(n) < ε