Abstract
Les fonctions arithmétiques associées aux systèmes de représentations d’entiers, comme le développement dans une base donnée, satisfont généralement des relations de récurrence qui facilitent considérablement l’étude de leur valeur moyenne. Considérons par exemple la somme des chiffres en base 2, que nous désignons par σ(n).
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Bibliographie
J.-P. Allouche & J. Shallit, The ring of k-regular sequences, Theor. Comp. Sci. 98 (1992), 163–187.
J. Brillhart, P. Erdős & P. Morton, On sums of Rudin-Shapiro coefficients II, Pac. J. Math. 107 (1983), 39–69.
J. Brillhart & P. Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, III. J. Math. 22 (1978), 126–148.
L. E. Bush, An asymptotic formula for the average sums of the digits of integers, Amer. Math. Monthly 47 (1940), 154–156.
E. Cateland, Suites digitales et suites k régulières, Thèse, Université de Bordeaux 1, 1992.
P. Cheo & S. Yien, A problem on the K-adic representation of positive integers, Acta Math. Sinica 5 (1955), 433–438.
J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983), 107–115.
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), 161–176.
J. Coquet & P. van den Bosch, A summation formula involving Fibonacci digits, J. Number Theory 22 (1986), 139–146.
H. Delange, Sur la fonction sommatoire de la fonction “somme des chiffres”, Ens. Math. 21 (1975), 31–47.
J.-M. Dumont, Formules sommatoires et systèmes de numération liés aux substitutions, Séminaire de théorie des nombres de Bordeaux (1987/88), Exposé n° 39.
J.-M. Dumont & A. Thomas, Systèmes de numération et fonctions fractal es relatifs aux substitutions, Theor. Comp. Sci. 65 (1989), 153–169.
J.-M. Dumont & A. Thomas, Digital sum problems and substitutions on a finite alphabet, J. Number Theory 39 (1991), 351–366.
P. Flajolet & L. Ramshaw, A note on Gray code and odd-even merge, SIAM J. Comp. 9 (1980), 142–158.
P. Flajolet, P. Grabner, P. Kirschenhoffer, H. Prodinger & R. Tichy, Mellin transforms and asymptotics: digital sums. Theoret. Comput Sci. 123 (1994), no. 2, 291–314.
D. M. Foster, Estimates for a remainder term associated with the sum of digits function, Glasgow Math. J. 29 (1987), 109–129.
P. J. Grabner & R. F. Tichy, Contributions to digit expansions with respect to linear recurrences, J. Number Theory 36 (1990), 160–169.
H. Harboth, Number of odd binomial coefficients, Proc. Amer. Math. Soc. 63 (1977), 19–22.
J. Honkala, On number systems with negative digits, Ann. Acad. Sci. Fenn Series A. I. Mathematica 14 (1989), 149–156.
R. E. Kennedy & C. N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Quarterly 29 (1991), 145–149.
P. Kirschenhoffer, Subblock occurrences in the q-ary representation of n, Siam J. Alg. Disc. Meth. 4 (1983), 231–236.
P. Kirschenhoffer & H. Prodinger, Subblock occurrences in positional number systems and Gray code representation, J. Inf. Opt Sci. 5 (1984), 29–42.
P. Kirschenhoffer & R. F. Tichy, On the distribution of digits in Cantor representations of integers, J. Number Theory 18 (1984), 121–134.
G. Larcher & R. F. Tichy, Some number-theoretical properties of generalized sum-of-digit functions, Acta Arith. 52 (1989) 183–196
M. D. Mac Ilroy, The number of l’s in binary integers: bounds and extremal properties, SIAM J. Comput. 3 (1974), 225–261.
L. Mirsky, A theorem on representations of integers in the scale of r, Scripta Math. 15 (1949), 11–12.
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969), 719–721.
A. Pethö & R. F. Tichy, On digit expansions with respect to linear recurrences, J. Number Theory 33 (1989), 243–256.
H. Prodinger, Generalizing the “sum of digits” function, SIAM J. Alg. Disc. Meth. 3 (1982), 35–42.
P. Shiu & A. H. Osbaldestin, A correlated digital sum problem associated with sums of three squares, Bull. London Math. Soc. 21 (1989), 369–374.
A. H. Stein, Exponential sums related to binomial coefficient parity, Proc. Amer. Math. Soc. 80 (1980), 526–530.
A. H. Stein, Exponential sums of an iterate of the binary sum of digit function, Indiana Univ. Math. J. 31 (1982), 309–315.
A. H. Stein, Exponential sums of sum-of-digit functions, III. J. Math. 30 (1986), 660–675.
A. H. Stein, Exponential sums of digit counting functions, in: J.M. de Konincket C. LeVesque (eds.), Théorie des Nombres (Québec 5–18/7/87), 861–868, Walter de Gruyter, Berlin-New York, 1989.
K. B. Stolarsky, Digital sums and binomial coefficients, Notices Amer. Math. Soc. 22 (1975), A 669, Abstract # 728-A7.
K. B. Stolarski, Power and exponential sums related to binomial digit parity, SIAM J. Appl Math. 32 (1977), 717–730.
J. R. Trollope, Generalized bases and digital stuns, ;Amer. Math. Monthly 74 (1967), 690–694.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 (1967), 21–25.
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Tenenbaum, G. (1997). Sur La Non-Dérivabilité de Fonctions Périodiques Associées à Certaines Formules Sommatoires. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös I. Algorithms and Combinatorics, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60408-9_10
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