# Sums of the Digits in Bases 2 and 3

• Jean-Marc Deshouillers
• Laurent Habsieger
• Shanta Laishram
• Bernard Landreau
Chapter

## 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
$$\displaystyle{\mathop{\mathrm{Card}}\nolimits \{n \leq N: \left \vert s_{3}(n) - s_{2}(n)\right \vert \leq 0.1457205\log n\}\,>\, N^{0.970359}.}$$
The proof only uses the separate (or marginal) distributions of the values of s2(n) and s3(n).

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## Notes

### Acknowledgements

The authors are indebted to Bernard Bercu for several discussions on the notion of “spacing” between two random variables, a notion to be developed later. They also thank the Referees for their constructive comments. The first, third, and fourth authors wish to thank the Indo-French centre CEFIPRA for the support permitting them to collaborate on this project (ref. 5401-A). The first named author acknowledges with thank the support of the French-Austrian project MuDeRa (ANR and FWF).

### References

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N.L. Bassily, I. Kátai, Distribution of the values of q-additive functions, on polynomial sequences. Acta Math. Hungar. 68, 353–361 (1995)
2. 2.
J.-M. Deshouillers, A footnote to The least non zero digit of n! in base 12. Unif. Distrib. Theory 7, 71–73 (2012)Google Scholar
3. 3.
J.-M. Deshouillers, I. Ruzsa, The least non zero digit of n! in base 12. Publ. Math. Debr. 79, 395–400 (2011)
4. 4.
M. Drmota, The joint distribution of q-additive functions. Acta Arith. 100, 17–39 (2001)
5. 5.
C. Stewart, On the representation of an integer in two different bases. J. Reine Angew. Math. 319, 63–72 (1980)

© Springer International Publishing AG 2017

## Authors and Affiliations

• Jean-Marc Deshouillers
• 1
• Laurent Habsieger
• 2
• Shanta Laishram
• 3
• Bernard Landreau
• 4
1. 1.Institut Mathématique de Bordeaux, UMR 5251Bordeaux INP, Université de Bordeaux, CNRSTalenceFrance
2. 2.Institut Camille JordanUniversité de Lyon, CNRS UMR 5208Villeurbanne CedexFrance
3. 3.Indian Statistical InstituteNew DelhiIndia
4. 4.LAREMA Laboratoire Angevin de REcherche en MAthématiques, UMR 6093, FR 2962Université d’Angers, CNRSAngersFrance