On a conjecture of Dekking : The sum of digits of even numbers

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

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 17-24
  • ISSN: 1246-7405

Abstract

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Let q ≥ 2 and denote by s q the sum-of-digits function in base q . For j = 0 , 1 , ⋯ , q - 1 consider # { 0 ≤ n < N : s q ( 2 n ) ≡ j ( mod q ) } . 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.

How to cite

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Boreico, Iurie, El-Baz, Daniel, and Stoll, Thomas. "On a conjecture of Dekking : The sum of digits of even numbers." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 17-24. <http://eudml.org/doc/275820>.

@article{Boreico2014,
abstract = {Let $q\ge 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,\dots ,q-1$ consider\[\# \lbrace 0 \le n &lt; N : \;\;s\_q(2n) \equiv j \pmod \{q\} \rbrace .\]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.},
affiliation = {Department of Mathematics Stanford University 450 Serra Mall Stanford, California 94305, USA; School of Mathematics University of Bristol University Walk Bristol, BS8 1TW, United Kingdom; 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},
author = {Boreico, Iurie, El-Baz, Daniel, Stoll, Thomas},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Dekking's conjecture; sum of digits; Newman phenomenon},
language = {eng},
month = {4},
number = {1},
pages = {17-24},
publisher = {Société Arithmétique de Bordeaux},
title = {On a conjecture of Dekking : The sum of digits of even numbers},
url = {http://eudml.org/doc/275820},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Boreico, Iurie
AU - El-Baz, Daniel
AU - Stoll, Thomas
TI - On a conjecture of Dekking : The sum of digits of even numbers
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 17
EP - 24
AB - Let $q\ge 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,\dots ,q-1$ consider\[\# \lbrace 0 \le n &lt; N : \;\;s_q(2n) \equiv j \pmod {q} \rbrace .\]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.
LA - eng
KW - Dekking's conjecture; sum of digits; Newman phenomenon
UR - http://eudml.org/doc/275820
ER -

References

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  2. J. Coquet, A summation formula related to the binary digits. Invent. Math. 73 (1983), 107–115. Zbl0528.10006MR707350
  3. F. M. Dekking, On the distribution of digits in arithmetic sequences. Séminaire de Théorie des Nombres de Bordeaux, exposé no.32 (1983). Zbl0529.10047MR750333
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  8. V. Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int. J. Math. Math. Sci., ID 908045 (2008). Zbl1247.11122MR2448274
  9. V. Shevelev, Exact exponent in the remainder term of Gelfond’s digit theorem in the binary case, Acta Arith. 136 (2009), 91–100. Zbl1232.11012MR2469946
  10. I. Shparlinski, On the size of the Gelfond exponent, J. Number Theory 130, (4) (2010), 1056–1060. Zbl1209.11008MR2600421
  11. G. Tenenbaum, Sur la non-dérivabilité de fonctions périodiques associées à certaines fonctions sommatoires, in: R.L. Graham & J. Nesetril (eds), The mathematics of Paul Erdős, Algorithms and Combinatorics 13 Springer Verlag, (1997), 117–128. Zbl0869.11019MR1425180

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