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 → ∞ 1 N ∑ n ≤ N s 2 ( n r ) s 2 ( n ) ≤ r . In this paper we prove this conjecture. We show that for polynomials P 1 ( X ) , P 2 ( X ) ∈ Z [ X ] of degrees r 1 , r 2 ≥ 1 and integers q 1 , q 2 ≥ 2, we have lim N → ∞ 1 N ∑ n ≤ N s q 1 ( P 1 ( n ) ) s q 2 ( P 2 ( n ) ) = r 1 ( q 1 - 1 ) log q 2 r 2 ( q 2 - 1 ) log q 1 . We also present a variant of the problem to polynomial values of prime numbers.