On a second conjecture of Stolarsky: the sum of digits of polynomial values
Résumé
Let $q,r\geq 2$ 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 $r_1, r_2\geq 1$ and integers $q_1, q_2\geq 2$ we have $$\lim_{N\to \infty} \frac{1}{N}\sum_{n\leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))}=\frac{r_1 (q_1-1)\log q_2}{r_2(q_2-1) \log q_1}.$$ We also present a variant of the problem to polynomial values of prime numbers.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)
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