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 lim inf n→∞ s 2 (n 2 ) s 2 (n) = 0. He conjectured that, just 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 polynomialwith 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.

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Let q, m ≥ 2 be integers with (m, q − 1) = 1. Denote by sq(n) the sum of digits of n in the q-ary digital expansion. Further let p(x) ∈ Z[x] be a polynomial of degree h ≥ 3 with p(N) ⊂ N. We show that there exist C = C(q, m, p) > 0 and N 0 = N 0 (q, m, p) ≥ 1, such that for all g ∈ Z and all N ≥ N 0 , #{0 ≤ n < N : sq(p(n)) ≡ g mod m} ≥ CN 4/(3h+1) . This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is CN 2/h! .

Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s 2 (n) = s 2 (n 2 ). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n 2 , looking at the subsets of n where s q (n) = s q (n 2 ) = k for fixed k.