On the distribution of complex-valued multiplicative functions

Antanas Laurinčikas

Antanas Laurinčikas

Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 1, pp. 183-203

Abstract (VO)
Résumé

Let $g_{j} (m), j = 1, 2$ , be complex-valued multiplicative functions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measure

\frac{1}{n} card \{0 \leq m \leq n : (g_{1} (m), g_{2} (m)) \in A\}, A \in ℬ (ℂ^{2}),

as

n \to \infty

.

Soient $g_{1} (m), g_{2} (m)$ deux fonctions multiplicatives a valeurs complexes. Dans un article on donne les conditions nécessaires et suffisantes de la convergence en un certain sens de

\frac{1}{n} N_{n} ((g_{1} (m), g_{2} (m)) \in A), A \in ℬ (ℂ^{2}),

quand

n \to \infty .

MR Zbl

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     author = {Antanas Laurin\v{c}ikas},
     title = {On the distribution of complex-valued multiplicative functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {183--203},
     year = {1996},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {1},
     zbl = {0864.11039},
     mrnumber = {1399954},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1996__8_1_183_0/}
}

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Antanas Laurinčikas. On the distribution of complex-valued multiplicative functions. Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 1, pp. 183-203. https://jtnb.centre-mersenne.org/item/JTNB_1996__8_1_183_0/

Bibliographie
Cité par

[1] H. Delange, Sur la distribution des valeurs des functions multiplicatives complexes, C. R. Acad. Sc. Paris, 276, Série A (1973), pp. 161-164. | Zbl | MR

[2] J. Kubilius, Probabilistic Methods in the Theory of Numbers, Amer. Math. Soc. Translations of Math. Monographs, No 11, Providence, 1964. | Zbl | MR

[3] M.I. Tuliaganova, Multidimensional distributions for additive functions with the unit normalization, Izv.AN Uz. SSR, ser. phys.-mat., No 2 (1974), pp. 29-35 (in Russian).

[4] A. Laurincikas, Multidimensional distribution of values of multiplicative functions, Liet. matem. rink., 15, No 2 (1975), pp. 13-24 (in Russian). | Zbl | MR

[5] R. Sliesoraitiens, Application of the Cramér-Wold method for multidimensional distributions of additive arithmetical functions, Liet. matem. rink., 16, No 2 (1976), pp. 155-172 (in Russian). | MR

[6] A. Laurinčikas, On joint distribution of values of arithmetical functions, Liet. matem. rink., 31, No 3 (1991), pp. 433-454 (in Russian). | Zbl | MR

[7] A. Laurinčikas, On probability measures on the multidemensional complex plane, Liet. matem. rink., 34, No 3 (1994), pp. 331-346 (in Russian). | Zbl | MR

[8] H. Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. (3), 78 (1961), pp. 273-304. | Zbl | MR | Numdam

[9] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretisher Funktionen, Acta Math. Acad. Sci. Hung., 19 (1968), pp. 365-403. | Zbl | MR

[10] H. Delange, On the distribution modulo I of additive functions, J. Indian Math. Soc., 34 (1970), pp. 215-235. | Zbl | MR

[11] P.D.T.A. Elliott, Probabilistic Number Theory I, Springer-Verlag, 1979. | Zbl | MR

[12] J. Kubilius, Probabilistic methods in the theory of arithmetical functions, Akt. probl. analit. teorii tchisel, Minsk, 1974, pp. 81-118 (in Russian). | Zbl

[13] N M. Timofeev, B. V. Levin, Analytic method in probabilistic number theory, Utch. zap. Vladim. gos. ped.inst. mat., 57 (2) (1971), pp. 57-150 (in Russian). | MR

[14] H. Delange, A remark on multiplicative functions, Bull. London Math. Soc., 2 (1970), pp. 183-185. | Zbl | MR

[15] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publications de l'Institut Élie Cartan, Université de Nancy I, 1990. | Zbl

[16] B.V. Levin, N.M. Timofeev, S.T. Tuliaganov, Distribution of values of multiplicative functions, Liet. matem. rink., 13, No 1 (1973), pp. 87-100 (in Russian). | Zbl | MR