The distribution of the sum-of-digits function

Michael Drmota; Johannes Gajdosik

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 1, page 17-32
  • ISSN: 1246-7405

Abstract

top
By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.

How to cite

top

Drmota, Michael, and Gajdosik, Johannes. "The distribution of the sum-of-digits function." Journal de théorie des nombres de Bordeaux 10.1 (1998): 17-32. <http://eudml.org/doc/248163>.

@article{Drmota1998,
abstract = {By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.},
author = {Drmota, Michael, Gajdosik, Johannes},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {digital expansions; central limit theorem; sum-of-digits function; linear recurring base sequences; local limit law},
language = {eng},
number = {1},
pages = {17-32},
publisher = {Université Bordeaux I},
title = {The distribution of the sum-of-digits function},
url = {http://eudml.org/doc/248163},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Drmota, Michael
AU - Gajdosik, Johannes
TI - The distribution of the sum-of-digits function
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 1
SP - 17
EP - 32
AB - By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.
LA - eng
KW - digital expansions; central limit theorem; sum-of-digits function; linear recurring base sequences; local limit law
UR - http://eudml.org/doc/248163
ER -

References

top
  1. [1] N.L. Bassily and I. Kátai, Distribution of the values of q-additive functions on polynomial sequences, Acta Math. Hung.68 (1995), 353-361. Zbl0832.11035MR1333478
  2. [2] R. Bellman and H.N. Shapiro, On a problem in additive number theory, Ann. Math.49 (1948), 333-340. Zbl0031.25401MR23864
  3. [3] L.E. Bush, An asymptotic formula for the average sum of the digits of integers, Am. Math. Monthly47 (1940), 154-156. Zbl0025.10601MR1225JFM66.1212.01
  4. [4] J. Coquet, Power sums of digital sums, J. Number Th.22 (1986), 161-176. Zbl0578.10009MR826949
  5. [5] H. DelangeSur la fonction sommatoire de la fonction "Somme de Chiffres", L 'Enseignement math.21 (1975), 31-77. Zbl0306.10005MR379414
  6. [6] M. Drmota and M. Skalba, The parity of the Zeckendorf sum-of-digits-function, preprint. Zbl0958.11015MR1751039
  7. [7] J.M. Dumont and A. Thomas, Digital sum moments and substitutions, Acta Arith.64 (1993), 205-225. Zbl0774.11041MR1225425
  8. [8] J.M. Dumont and A. Thomas, Gaussian asymptotic properties of the sum-of-digits functions, J. Number Th.62 (1997), 19-38. Zbl0869.11009MR1430000
  9. [9] C.-G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math.77 (1945), 1-125. Zbl0060.28705MR14626
  10. [10] J. Gajdosik, Kombinatorische Faktorisierungen und Ziffernentwicklungen, thesis, TU Wien, 1996. 
  11. [11] P.J. Grabner, P. Kirschenhofer, H. Prodinger, and R.F. Tichy, On the moments of the sum-of-digits function, in: Applications of Fibonacci Numbers5 (1993), 263-271 Zbl0797.11012MR1271366
  12. [12] P.J. Grabner and R.F. Tichy, Contributions to digit expansions with respect to linear recurrences, J. Number Th.36 (1990), 160-169. Zbl0711.11004MR1072462
  13. [13] P. Grabner and R.F. Tichy, a-Expansions, linear recurrences, and the sum-of-digits function, manuscripta math.70 (1991), 311-324. Zbl0725.11005MR1089067
  14. [14] R.E. Kennedy and C.N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Q.29 (1991), 145-149. Zbl0728.11004MR1119401
  15. [15] P. Kirschenhofer, On the variance of the sum of digits function, Lecture Notes Math.1452 (1990), 112-116. Zbl0714.11005MR1084640
  16. [16] W. Parry, On the,β-expansion of real numbers, Acta Math. Acad. Sci. Hung., 12 (1961), 401-416. Zbl0099.28103
  17. [17] A. Pethö and R.F. Tichy, On digit expansions with respect to linear recurrences, J. Number Th.33 (1989), 243-256. Zbl0676.10010MR1034204
  18. [18] J. Schmid, The joint distribution of the binary digits of integer multiples, Acta Arith.43 (1984), 391-415. Zbl0489.10008MR756290
  19. [19] W.M. Schmidt, The joint distribution the digits of certain integer s-tuples, Studies in pure mathematics, Mem. of P. Turan (1983), 605-622. Zbl0523.10030MR820255
  20. [20] H. Trollope, An explicit expression for binary digital sums, Meth. Mag.41 (1968), 21-25. Zbl0162.06303MR233763

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.