The distribution of the sum-of-digits function
Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 17-32.

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.

Dans cet article, nous démontrons que la fonction “somme de chiffres” relative à des recurrences linéaires finies et infinies paxticulieres) satisfait à un theoreme central limite. Nous obtenons aussi un théorème limite local.

Keywords: digital expansions, central limit theorem
@article{JTNB_1998__10_1_17_0,
     author = {Drmota, Michael and Gajdosik, Johannes},
     title = {The distribution of the sum-of-digits function},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {17--32},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {1},
     year = {1998},
     mrnumber = {1827283},
     zbl = {0916.11049},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_1998__10_1_17_0/}
}
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
SP  - 17
EP  - 32
VL  - 10
IS  - 1
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_1998__10_1_17_0/
LA  - en
ID  - JTNB_1998__10_1_17_0
ER  - 
%0 Journal Article
%A Drmota, Michael
%A Gajdosik, Johannes
%T The distribution of the sum-of-digits function
%J Journal de théorie des nombres de Bordeaux
%D 1998
%P 17-32
%V 10
%N 1
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_1998__10_1_17_0/
%G en
%F JTNB_1998__10_1_17_0
Drmota, Michael; Gajdosik, Johannes. The distribution of the sum-of-digits function. Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 17-32. http://archive.numdam.org/item/JTNB_1998__10_1_17_0/

[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. | MR | Zbl

[2] R. Bellman and H.N. Shapiro, On a problem in additive number theory, Ann. Math. 49 (1948), 333-340. | MR | Zbl

[3] L.E. Bush, An asymptotic formula for the average sum of the digits of integers, Am. Math. Monthly 47 (1940), 154-156. | JFM | MR | Zbl

[4] J. Coquet, Power sums of digital sums, J. Number Th. 22 (1986), 161-176. | MR | Zbl

[5] H. Delange Sur la fonction sommatoire de la fonction "Somme de Chiffres", L 'Enseignement math. 21 (1975), 31-77. | MR | Zbl

[6] M. Drmota and M. Skalba, The parity of the Zeckendorf sum-of-digits-function, preprint. | MR

[7] J.M. Dumont and A. Thomas, Digital sum moments and substitutions, Acta Arith. 64 (1993), 205-225. | MR | Zbl

[8] J.M. Dumont and A. Thomas, Gaussian asymptotic properties of the sum-of-digits functions, J. Number Th. 62 (1997), 19-38. | MR | Zbl

[9] C.-G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1-125. | MR | Zbl

[10] J. Gajdosik, Kombinatorische Faktorisierungen und Ziffernentwicklungen, thesis, TU Wien, 1996.

[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 Numbers 5 (1993), 263-271 | MR | Zbl

[12] P.J. Grabner and R.F. Tichy, Contributions to digit expansions with respect to linear recurrences, J. Number Th. 36 (1990), 160-169. | MR | Zbl

[13] P. Grabner and R.F. Tichy, a-Expansions, linear recurrences, and the sum-of-digits function, manuscripta math. 70 (1991), 311-324. | MR | Zbl

[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. | MR | Zbl

[15] P. Kirschenhofer, On the variance of the sum of digits function, Lecture Notes Math. 1452 (1990), 112-116. | MR | Zbl

[16] W. Parry, On the,β-expansion of real numbers, Acta Math. Acad. Sci. Hung., 12 (1961), 401-416. | Zbl

[17] A. Pethö and R.F. Tichy, On digit expansions with respect to linear recurrences, J. Number Th. 33 (1989), 243-256. | MR | Zbl

[18] J. Schmid, The joint distribution of the binary digits of integer multiples, Acta Arith. 43 (1984), 391-415. | MR | Zbl

[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. | MR | Zbl

[20] H. Trollope, An explicit expression for binary digital sums, Meth. Mag. 41 (1968), 21-25. | MR | Zbl