On a conjecture of Dekking : The sum of digits of even numbers
Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 17-24.

Let q2 and denote by s q the sum-of-digits function in base q. For j=0,1,,q-1 consider

#{0n<N:sq(2n)j(modq)}.

In 1983, F. M. Dekking conjectured that this quantity is greater than N/q and, respectively, less than N/q for infinitely many N, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

A propos d’une conjecture de Dekking : la somme des chiffres des nombres pairs

Soient q2 et s q la fonction somme des chiffres en base q. Pour j=0,1,...,q-1 on considère

#{0n<N:sq(2n)j(modq)}.

En 1983, F. M. Dekking a conjecturé que cette quantité est strictement supérieure à N/q et, respectivement, strictement inférieure à N/q pour une infinité de N, affirmant ce faisant l’absence d’un phénomène de dérive (ou phénomène de Newman). Dans cet article, nous démontrons sa conjecture.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.856
Iurie Boreico 1; Daniel El-Baz 2; Thomas Stoll 3

1 Department of Mathematics Stanford University 450 Serra Mall Stanford, California 94305, USA
2 School of Mathematics University of Bristol University Walk Bristol, BS8 1TW, United Kingdom
3 1. Université de Lorraine Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France 2. CNRS Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France
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Iurie Boreico; Daniel El-Baz; Thomas Stoll. On a conjecture of Dekking : The sum of digits  of even numbers. Journal de Théorie des Nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 17-24. doi : 10.5802/jtnb.856. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.856/

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