On the distribution of complex-valued multiplicative functions

Antanas Laurinčikas

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 1, page 183-203
  • ISSN: 1246-7405

Abstract

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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 1 n card 0 ≤ m ≤ n : ( g 1 ( m ) , g 2 ( m ) ) ∈ A , A ∈ ℬ ( ℂ 2 ) , as n → ∞ .

How to cite

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Laurinčikas, Antanas. "On the distribution of complex-valued multiplicative functions." Journal de théorie des nombres de Bordeaux 8.1 (1996): 183-203. <http://eudml.org/doc/247831>.

@article{Laurinčikas1996,
abstract = {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\begin\{equation*\}\frac\{1\}\{n\} \text\{ card\} \left\lbrace 0 \le m \le n : (g\_1(m), g\_2(m)) \in A\right\rbrace , A \in \mathcal \{B\}(\mathbb \{C\}^2), \end\{equation*\}as $n \rightarrow \infty $.},
author = {Laurinčikas, Antanas},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {multiplicative functions; joint limiting distribution},
language = {eng},
number = {1},
pages = {183-203},
publisher = {Université Bordeaux I},
title = {On the distribution of complex-valued multiplicative functions},
url = {http://eudml.org/doc/247831},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Laurinčikas, Antanas
TI - On the distribution of complex-valued multiplicative functions
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 1
SP - 183
EP - 203
AB - 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\begin{equation*}\frac{1}{n} \text{ card} \left\lbrace 0 \le m \le n : (g_1(m), g_2(m)) \in A\right\rbrace , A \in \mathcal {B}(\mathbb {C}^2), \end{equation*}as $n \rightarrow \infty $.
LA - eng
KW - multiplicative functions; joint limiting distribution
UR - http://eudml.org/doc/247831
ER -

References

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  1. [1] H. Delange, Sur la distribution des valeurs des functions multiplicatives complexes, C. R. Acad. Sc. Paris, 276, Série A (1973), pp. 161-164. Zbl0245.10043MR369302
  2. [2] J. Kubilius, Probabilistic Methods in the Theory of Numbers, Amer. Math. Soc. Translations of Math. Monographs, No 11, Providence, 1964. Zbl0133.30203MR160745
  3. [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. [4] A. Laurincikas, Multidimensional distribution of values of multiplicative functions, Liet. matem. rink., 15, No 2 (1975), pp. 13-24 (in Russian). Zbl0324.60016MR384737
  5. [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). MR432580
  6. [6] A. Laurinčikas, On joint distribution of values of arithmetical functions, Liet. matem. rink., 31, No 3 (1991), pp. 433-454 (in Russian). Zbl0732.11037MR1162237
  7. [7] A. Laurinčikas, On probability measures on the multidemensional complex plane, Liet. matem. rink., 34, No 3 (1994), pp. 331-346 (in Russian). Zbl0831.60006MR1355242
  8. [8] H. Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. (3), 78 (1961), pp. 273-304. Zbl0234.10043MR169829
  9. [9] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretisher Funktionen, Acta Math. Acad. Sci. Hung., 19 (1968), pp. 365-403. Zbl0165.05804MR230694
  10. [10] H. Delange, On the distribution modulo I of additive functions, J. Indian Math. Soc., 34 (1970), pp. 215-235. Zbl0234.10044MR491576
  11. [11] P.D.T.A. Elliott, Probabilistic Number Theory I, Springer-Verlag, 1979. Zbl0431.10029MR551361
  12. [12] J. Kubilius, Probabilistic methods in the theory of arithmetical functions, Akt. probl. analit. teorii tchisel, Minsk, 1974, pp. 81-118 (in Russian). Zbl0333.10034
  13. [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). MR302596
  14. [14] H. Delange, A remark on multiplicative functions, Bull. London Math. Soc., 2 (1970), pp. 183-185. Zbl0212.39602MR262191
  15. [15] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publications de l'Institut Élie Cartan, Université de Nancy I, 1990. Zbl0788.11001
  16. [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). Zbl0257.10024MR314790

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