The distribution of the sum-of-digits function
Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 1, pp. 17-32.

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.

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.

DOI : https://doi.org/10.5802/jtnb.216
Mots clés : 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},
     doi = {10.5802/jtnb.216},
     zbl = {0916.11049},
     mrnumber = {1827283},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.216/}
}
Michael Drmota; Johannes Gajdosik. The distribution of the sum-of-digits function. Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 1, pp. 17-32. doi : 10.5802/jtnb.216. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.216/

[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 1333478 | Zbl 0832.11035

[2] R. Bellman and H.N. Shapiro, On a problem in additive number theory, Ann. Math. 49 (1948), 333-340. | MR 23864 | Zbl 0031.25401

[3] L.E. Bush, An asymptotic formula for the average sum of the digits of integers, Am. Math. Monthly 47 (1940), 154-156. | JFM 66.1212.01 | MR 1225 | Zbl 0025.10601

[4] J. Coquet, Power sums of digital sums, J. Number Th. 22 (1986), 161-176. | MR 826949 | Zbl 0578.10009

[5] H. Delange Sur la fonction sommatoire de la fonction "Somme de Chiffres", L 'Enseignement math. 21 (1975), 31-77. | MR 379414 | Zbl 0306.10005

[6] M. Drmota and M. Skalba, The parity of the Zeckendorf sum-of-digits-function, preprint. | MR 1751039

[7] J.M. Dumont and A. Thomas, Digital sum moments and substitutions, Acta Arith. 64 (1993), 205-225. | MR 1225425 | Zbl 0774.11041

[8] J.M. Dumont and A. Thomas, Gaussian asymptotic properties of the sum-of-digits functions, J. Number Th. 62 (1997), 19-38. | MR 1430000 | Zbl 0869.11009

[9] C.-G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Math. 77 (1945), 1-125. | MR 14626 | Zbl 0060.28705

[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 1271366 | Zbl 0797.11012

[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 1072462 | Zbl 0711.11004

[13] P. Grabner and R.F. Tichy, a-Expansions, linear recurrences, and the sum-of-digits function, manuscripta math. 70 (1991), 311-324. | MR 1089067 | Zbl 0725.11005

[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 1119401 | Zbl 0728.11004

[15] P. Kirschenhofer, On the variance of the sum of digits function, Lecture Notes Math. 1452 (1990), 112-116. | MR 1084640 | Zbl 0714.11005

[16] W. Parry, On the,β-expansion of real numbers, Acta Math. Acad. Sci. Hung., 12 (1961), 401-416. | Zbl 0099.28103

[17] A. Pethö and R.F. Tichy, On digit expansions with respect to linear recurrences, J. Number Th. 33 (1989), 243-256. | MR 1034204 | Zbl 0676.10010

[18] J. Schmid, The joint distribution of the binary digits of integer multiples, Acta Arith. 43 (1984), 391-415. | MR 756290 | Zbl 0489.10008

[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 820255 | Zbl 0523.10030

[20] H. Trollope, An explicit expression for binary digital sums, Meth. Mag. 41 (1968), 21-25. | MR 233763 | Zbl 0162.06303