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Sums of the Digits in Bases 2 and 3

In: Number Theory – Diophantine Problems, Uniform Distribution and Applications

Author

Listed:
  • Jean-Marc Deshouillers

    (Bordeaux INP, Université de Bordeaux, CNRS, Institut Mathématique de Bordeaux, UMR 5251)

  • Laurent Habsieger

    (Université de Lyon, CNRS UMR 5208, Institut Camille Jordan)

  • Shanta Laishram

    (Indian Statistical Institute)

  • Bernard Landreau

    (Université d’Angers, CNRS, LAREMA Laboratoire Angevin de REcherche en MAthématiques, UMR 6093, FR 2962)

Abstract

Let b ≥ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N, one has Card { n ≤ N : s 3 ( n ) − s 2 ( n ) ≤ 0 . 1457205 log n } > N 0 . 970359 . $$\displaystyle{\mathop{\mathrm{Card}}\nolimits \{n \leq N: \left \vert s_{3}(n) - s_{2}(n)\right \vert \leq 0.1457205\log n\}\,>\, N^{0.970359}.}$$ The proof only uses the separate (or marginal) distributions of the values of s 2(n) and s 3(n).

Suggested Citation

  • Jean-Marc Deshouillers & Laurent Habsieger & Shanta Laishram & Bernard Landreau, 2017. "Sums of the Digits in Bases 2 and 3," Springer Books, in: Christian Elsholtz & Peter Grabner (ed.), Number Theory – Diophantine Problems, Uniform Distribution and Applications, pages 211-217, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-55357-3_9
    DOI: 10.1007/978-3-319-55357-3_9
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