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On the distribution of \(q\)-additive functions. (Sur la répartition des fonctions \(q\)-additives.) (French) Zbl 0788.11032

Let \(f\) be a \(q\)-additive function with values in a metrizable locally compact commutative group \(G\). A necessary and sufficient condition is found for \(f\) to have a limiting distribution with a suitable centering; in particular, it is proved that for a compact group, such a centering always exists. Also a necessary and sufficient condition is found for the continuity of the limiting distribution when it exists.

MSC:

11K65 Arithmetic functions in probabilistic number theory
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References:

[1] Bourbaki, N., Topologie générale, deuxième édition, 1958, Hermann, Paris.
[2] Coquet, J., Sur la mesure spectrale des suites q-multiplicatives, Ann. Inst. Fourier(Grenoble) 29 (1979), 163-170. · Zbl 0386.10031
[3] Coquet, J., Répartition modulo 1 des suites q-additives, Comment. Math.21 (1980), 23-42. · Zbl 0434.10030
[4] Coquet, J., Kamae, T., Mendès France, M., Sur la mesure spectrale de certaines suites arithmétiques, Bull. Soc. math. France105 (1977), 369-384. · Zbl 0383.10035
[5] Delange, H., Sur les fonctions q-additives ou q-multiplicatives, Acta Arithmetica21 (1972), 285-298. · Zbl 0219.10062
[6] Gelfond, A.O., Sur les nombres qui ont des propriétés additives ou multiplicatives données, Acta Arithmetica13 (1968), 259-265. · Zbl 0155.09003
[7] Kamae, T., Spectral properties of arithmetic functions, Séminaire Delange-Pisot-Poitou, (18e année, 1976/1977), fasc. 1, exp. 12, 8 pages (1977). · Zbl 0371.10044
[8] Legendre, A.M., Théorie des Nombres, éd. 2, 1808. · JFM 30.0201.03
[9] Lévy, P., L’addition des variables aléatoires définies sur une circonférence, Bull. Soc. math. France67 (1939), 1-41. · JFM 65.1346.01
[10] Loeve, M., Probability Theory I, 4th ed. 1977, Springer, New-York. · Zbl 0359.60001
[11] Mauclaire, J.-L., Distribution of the values of an additive arithmetical function with values in a locally compact metrizable abelian group, to appear. · Zbl 0936.11047
[12] Mendès France, M., Les suites à spectre vide et la répartition modulo 1, J. Number Theory5 (1973), 1-15. · Zbl 0252.10033
[13] Queffélec, M., Mesures spectrales associées à certaines suites arithmétiques, Bull. Soc. math. France107 (1979), 385-421. · Zbl 0435.42007
[14] Reiter, H., Classical hormone analysis and locally compact groups, Oxford, Clarendon Press, 1968. · Zbl 0165.15601
[15] Tortrat, A., Calcul des probabilités et introduction aux processus aléatoires, Mas-son (1971). · Zbl 0212.49201
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