Literature Cited
R. A. Aleksandryan and É. A. Mirzakhanyan, General Topology [in Russian], Vysshaya Shokola, Moscow (1979).
R. S. Baibulatov, Selected Questions of Probabilistic Number Theory [in Russian], Perm'skii GPI, Bashkirskii GPI, Perm (1976).
R. S. Baibulatov, Questions of Probabilistic Number Theory on Semigroups [in Russian], Perm'skii GPI, Bashkirskii GPI, Perm (1977).
P. Billingsley, Convergence of Probability Measures, Wiley (1968).
B. M. Bredikhin, “Free real semigroups with power densities,” Mat. Sb.,46(88), No. 2, 143–158 (1958).
B. M. Bredikhin, “The remainder term in the asymptotic formula for υG(x),” Izv. Vyssh. Uchebn. Zaved., Mat., No. 6 (19), 40–49 (1960).
I. P. Kubilyus, Probabilistic Methods in Number Theory [in Russian], Gos. Izd. Polit. Nauchn. Lit. LitSSR, Vilnius (1962).
I. Kubilyus, “Probabilistic methods in the theory of the distribution of arithmetic functions,” in: Current Problems of Analytic Number Theory [in Russian], Nauka i Tekhnika, Minsk (1974), pp. 81–118.
B. V. Levin and N. M. Timofeev, “Distribution of values of arithmetic functions. II,” Acta Arithm.,33, No. 4, 327–351 (1977).
B. V. Levin and N. M. Timofeev, “Comparison theorem for multiplicative functions”, Acta Arithm.,42, No. 1, 21–47 (1982).
A. G. Postnikov, Introduction to Analytic Number Theory [in Russian], Nauka, Moscow (1971).
N. M. Timofeev, “Distribution of values of additive functions”, in: Third International Vilnius Conference on Probability Theory and Mathematical Statistics. Abstracts of Reports [in Russian], Vol. 2, Vilnius (1981), pp. 188–189.
N. M. Timofeev and S. T. Tulyaganov, “Distribution of values of additive functions with weight 1/n. I”, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, No. 4, 31–37 (1982).
P. L. Khenneken and A. Tortra, Probability Theory and Some of Its Applications [in Russian], Nauka, Moscow (1974).
Z. Yushkis, “A theorem of P. Erdos and A. Wintner on ordered semigroups with regular normalization,” Liet. Mat. Rinkinys,4, No. 3, 429–450 (1964).
Z. Yushikis, “Limit theorems for additive functions, defined on ordered semigroups with regular normalization,” Liet. Mat. Rinkinys,4, No. 4, 565–603 (1964).
P. D. T. A. Elliott, Probabilistic Number Theory. Vol. I, II, Springer-Verlag, Berlin-Heidelberg-New York (1979, 1980).
J. Knopfmacher, “Arithmetical properties of finite rings and algebras, and analytic number theory. I–VI,” J. Reine Angew. Math.,252, 16–43 (1972); 254, 74–99, (1972);259, 157–170 (1973);270, 97–114 (1974);271, 95–121 (1974);277, 45–62 (1975).
J. Knopfmacher, “Fourier analysis of arithmetical functions,” Ann. Mat. Pura Appl.,109, 177–201 (1976).
B. V. Levin and N. M. Timofeev, “On the distribution of values of additive functions,” Acta Arithm.,26, No. 4, 333–364 (1975).
E. Marczewski, “On compact measures”, Fund. Math.,40, 113–124 (1953).
P. Nanopoulos, “Loi de Dirichlet sur N* et pseudoprobabilities”, C. R. Acad. Sci. Paris,280, Ser. A, No. 22, 1543–1546 (1975).
P. Nanopoulos, “Lois zeta et fonctions arithmetiques additives. Loi faible de grands nombres,” C. R. Acad. Sci. Paris,285, Ser. A, No. 13, 875–877 (1977).
E. M. Paul, “Density in the Light of probability theory. I, II, III,” Sankhya,A24, 103–114; 209–212 (1962); A25, 273–280 (1963).
W. Schwarz and J. Spilker, “Eine Anwendung des Approximationssatzes von Weierstrass-Stone auf Ramanujan-Summen”, Nieuw Archief voor Wiskunde (3),19, 198–209 (1971).
W. Schwarz and J. Spilker, “Mean values and Ramanujan expansions of almost even arithmetical functions”, in: Colloq. Math. Soc. Janos Bolyai, 13. Topics in Number Theory, Debrecen (Hungary), 1974 North-Holland, Amsterdam-Oxford-New York (1976), pp. 315–357.
F. Topsøe, “Preservation of weak convergence under mappings,” Ann. Math. Stat.,38, No. 6, 1661–1665 (1967).
H. Wegmann, “Beitrage zur Zahlentheorie auf freien Halbgruppen. I, II,” J. Reine Angew. Math.,221, 20–43; 150–159 (1966).
V. S. Varadarain, “Measures on topological spaces,” Mat. Sb.,55(97), No. 1, 35–100 (1961).
P. Billingsley, “An application of Prohorov's theorem to probabilistic number theory”, Ill. J. Math.,8, No. 4, 697–704 (1964).
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Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 25, No. 1, pp. 72–83, January–March, 1985.
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Kryžius, Z. Additive arithmetic functions on semigroups and the preservation of weak convergence of measures. Lith Math J 25, 35–43 (1985). https://doi.org/10.1007/BF00966295
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DOI: https://doi.org/10.1007/BF00966295