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

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.

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.

Reçu le :
Accepté le :
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DOI : https://doi.org/10.5802/jtnb.856
@article{JTNB_2014__26_1_17_0,
     author = {Iurie Boreico and Daniel El-Baz and Thomas Stoll},
     title = {On a conjecture of {Dekking} : {The} sum of digits  of even numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {17--24},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {1},
     year = {2014},
     doi = {10.5802/jtnb.856},
     zbl = {1300.11077},
     mrnumber = {3232764},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.856/}
}
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, Tome 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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