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On a second conjecture of Stolarsky: the sum of digits of polynomial values

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Abstract

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

$$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N} \frac{s_2(n^r)}{s_2(n)} \leq r.$$

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 ≥ 1 and integers q 1, q 2 ≥ 2, we have

$$\lim_{N \to \infty} \frac{1}{N} \sum_{n \leq N}\frac{s_{q_1}(P_1(n))}{s_{q_2}(P_2(n))} = \frac{r_1 (q_1 - 1) {\rm log}q_2}{r_2(q_2 - 1) {\rm log} q_1}.$$

We also present a variant of the problem to polynomial values of prime numbers.

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Authors and Affiliations

  1. UMR 7502, Institut Elie Cartan de Lorraine, Université de Lorraine, 54506, Vandoeuvre-lès-Nancy, France

    Manfred G. Madritsch & Thomas Stoll

  2. UMR 7502, Institut Elie Cartan de Lorraine, CNRS, 54506, Vandoeuvre-lès-Nancy, France

    Manfred G. Madritsch & Thomas Stoll

Authors
  1. Manfred G. Madritsch
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  2. Thomas Stoll
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Correspondence to Manfred G. Madritsch.

Additional information

The second author was supported by the Agence Nationale de la Recherche, Grant ANR-10-BLANC 0103 MUNUM.

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Madritsch, M.G., Stoll, T. On a second conjecture of Stolarsky: the sum of digits of polynomial values. Arch. Math. 102, 49–57 (2014). https://doi.org/10.1007/s00013-013-0587-z

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  • DOI: https://doi.org/10.1007/s00013-013-0587-z

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