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.
@article{JTNB_1998__10_1_17_0, author = {Michael Drmota and Johannes Gajdosik}, 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/} }
TY - JOUR TI - The distribution of the sum-of-digits function JO - Journal de Théorie des Nombres de Bordeaux PY - 1998 DA - 1998/// SP - 17 EP - 32 VL - 10 IS - 1 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.216/ UR - https://zbmath.org/?q=an%3A0916.11049 UR - https://www.ams.org/mathscinet-getitem?mr=1827283 UR - https://doi.org/10.5802/jtnb.216 DO - 10.5802/jtnb.216 LA - en ID - JTNB_1998__10_1_17_0 ER -
Michael Drmota; Johannes Gajdosik. The distribution of the sum-of-digits function. Journal de Théorie des Nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 17-32. doi : 10.5802/jtnb.216. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.216/
[1] Distribution of the values of q-additive functions on polynomial sequences, Acta Math. Hung. 68 (1995), 353-361. | MR: 1333478 | Zbl: 0832.11035
and ,[2] On a problem in additive number theory, Ann. Math. 49 (1948), 333-340. | MR: 23864 | Zbl: 0031.25401
and ,[3] 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] Power sums of digital sums, J. Number Th. 22 (1986), 161-176. | MR: 826949 | Zbl: 0578.10009
,[5] Sur la fonction sommatoire de la fonction "Somme de Chiffres", L 'Enseignement math. 21 (1975), 31-77. | MR: 379414 | Zbl: 0306.10005
[6] The parity of the Zeckendorf sum-of-digits-function, preprint. | MR: 1751039
and ,[7] Digital sum moments and substitutions, Acta Arith. 64 (1993), 205-225. | MR: 1225425 | Zbl: 0774.11041
and ,[8] Gaussian asymptotic properties of the sum-of-digits functions, J. Number Th. 62 (1997), 19-38. | MR: 1430000 | Zbl: 0869.11009
and ,[9] 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] Kombinatorische Faktorisierungen und Ziffernentwicklungen, thesis, TU Wien, 1996.
,[11] On the moments of the sum-of-digits function, in: Applications of Fibonacci Numbers 5 (1993), 263-271 | MR: 1271366 | Zbl: 0797.11012
, , , and ,[12] Contributions to digit expansions with respect to linear recurrences, J. Number Th. 36 (1990), 160-169. | MR: 1072462 | Zbl: 0711.11004
and ,[13] a-Expansions, linear recurrences, and the sum-of-digits function, manuscripta math. 70 (1991), 311-324. | MR: 1089067 | Zbl: 0725.11005
and ,[14] An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Q. 29 (1991), 145-149. | MR: 1119401 | Zbl: 0728.11004
and ,[15] On the variance of the sum of digits function, Lecture Notes Math. 1452 (1990), 112-116. | MR: 1084640 | Zbl: 0714.11005
,[16] On the,β-expansion of real numbers, Acta Math. Acad. Sci. Hung., 12 (1961), 401-416. | Zbl: 0099.28103
,[17] On digit expansions with respect to linear recurrences, J. Number Th. 33 (1989), 243-256. | MR: 1034204 | Zbl: 0676.10010
and ,[18] The joint distribution of the binary digits of integer multiples, Acta Arith. 43 (1984), 391-415. | MR: 756290 | Zbl: 0489.10008
,[19] 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] An explicit expression for binary digital sums, Meth. Mag. 41 (1968), 21-25. | MR: 233763 | Zbl: 0162.06303
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