Jonas Kubilius và nguồn gốc của lý thuyết số xác suất

Lithuanian Mathematical Journal - Tập 55 - Trang 25-47 - 2015
Vilius Stakėnas1
1Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania

Tóm tắt

Mục đích của khảo sát này là cung cấp một cái nhìn tổng quát về sự phát triển của Lý thuyết số xác suất, bắt đầu từ những khởi đầu cho đến những thập kỷ gần đây. Khác với việc pretender là đầy đủ, chúng tôi tập trung chủ yếu vào những ý tưởng và hiểu biết mà J. Kubilius đã đóng góp hoặc ảnh hưởng đến sự phát triển tiếp theo.

Từ khóa

#Lý thuyết số xác suất #Jonas Kubilius #sự phát triển #ý tưởng #hiểu biết

Tài liệu tham khảo

G.J. Babu, On the distribution of additive arithmetical functions of integral polynomials, Sankhyā, Ser. A, 34:323–334, 1972. G.J. Babu, A note on the invariance principle for additive functions, Sankhyā, Ser. A, 35:307–310, 1973. M.B. Barban and A.I. Vinogradov, On the number theoretic basis of probabilistic number theory, Dokl. Akad. Nauk SSSR, 154:495–496, 1964. L.E. Dickson, History of the Theory of Numbers, I., Carnegie Institution of Washington, Washington, 1919. P.D.T.A. Elliott, Probabilistic Number Theory I. Mean value theorems, Springer-Verlag, New York, Heidelberg, Berlin, 1979. P.D.T.A. Elliott, Probabilistic Number Theory II. Central Limit theorems, Springer-Verlag, New York, Heidelberg, Berlin, 1979. P.D.T.A. Elliott, Arithmetic Functions and Integer Products, Springer-Verlag, New York, Heidelberg, Berlin, 1985. P.D.T.A. Elliott, Duality in Analytic Number Theory, Cambridge Univ. Press, 1997. P. Erdős, On the density of some sequences of numbers. I, J. Lond. Math. Soc., 10:120–125, 1935. P. Erdős, On the density of some sequences of numbers. II, J. Lond. Math. Soc., 10:7–11, 1935. P. Erdős, On the density of some sequences of numbers. III, J. Lond. Math. Soc., 13:119–127, 1938. P. Erdős and M. Kac, The Gaussian law of errors in the theory of additive number-theoretic functions, Am. J. Math., 62:738–742, 1940. P. Erdős and A.Wintner, Additive arithmetical functions and statistical independence, Am. J. Math. Soc., 61:713–721, 1939. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math., 48:76–92, 1917. A. Hildebrand, The prime number theorem via the large sieve, Mathematika, 33:23–30, 1986. K.-H. Indlekofer, Über Verallgemeinerungen der Turán–Kubilius Ungleichung, Acta Arith., 60:67–73, 1989. K.-H. Indlekofer, A new method in probabilistic number theory, in Probability Theory and Applications, Mathematics and Its Applications, Vol. 80, Springer, 1992, pp. 299–308. A. Khinchine, Über einen Satz der Wahrscheinlichkeitsrechnung, Fundam. Math., 6:9–20, 1924. J. Knopfmacher and W.-B. Zhang, Number Theory Arising from Finite Fields, Marcel Dekker, New York, 2001. A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Heidelberg, 1933. J. Kubilius, Probabilistic Methods in the Theory of Numbers, Gosudarstv. Izdat. Politich. i Nauchn. Lit., Vilnius, 1962 (in Russian). English transl.: Transl. Math. Monogr., Vol. 11, Amer. Math. Soc., Providence, RI, 1964 (2nd ed., 1968; 3rd ed. with correct., 1978; 4th ed., 1992; 5th ed., 1997). J. Kubilius, Improvement of the estimation of the second central moment for additive arithmetical functions, Liet. Mat. Rink., 25(3):104–110, 1985. J. Kubilius, On some inequalities in the probabilistic number theory, in A. Laurinčikas et al. (Ed.), Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23–27 September 1996, VSP/TEV, Utrect/Vilnius, 1997, pp. 345–356. J.P. Kubilius and Yu. V. Linnik, Arithmetic modeling of Brownian motion, Izv. Vyssh. Uchebn. Zaved., Mat., 13(6):88–95, 1959. E. Landau, Sur quelques problèmes relatifs à la distribution des nombres premiers, Bull. Soc. Math. Fr., 28:25–38, 1900. J. Lee, The second central moment of additive functions, Proc. Am. Math. Soc., 114:104–110, 1992. W. J. LeVeque, On the size of certain number-theoretic functions, Trans. Am. Math. Soc., 66:440–463, 1949. B.V. Levin and N.M. Timofeev, Distribution of values of additive functions, Acta Arith., 33:327–351, 1977. A. Mačiulis, Optimal estimation of the rate of convergence in the central limit theorem for additive functions, Liet. Mat. Rink., 33(3):314–329, 1993. E. Manstavičius, Laws of the iterated logarithm for additive functions, Colloq. Math. Soc. Janos Bolyai, 51:279–299, 1987. E. Manstavičius, Probabilistic theory of additive functions related to systems of numeration, in A. Laurinčikas et al. (Eds.), Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23–27 September 1996, New Trends Probab. Stat., Vol. 4, VSP/TEV, Utrect/Vilnius, 1997, pp. 413–429. E. Manstavičius, Functional limit theorems in probabilistic number theory, in G. Halasz et al. (Eds.), Paul Erdős and His Mathematics. I, Bolyai Soc. Math. Stud., Vol. 11, Springer, Berlin, 2002, pp. 465–491. B. Martin and G. Tenenbaum, Sur l’inégalité de Turán–Kubilius friable, J. Reine Angew. Math., 647:175–234, 1998. J.-L. Mauclaire, Intégration et Théorie des Nombres, Hermann, Paris, 1997. E.V. Novosiolov, The new method in probabilistic number theory, Izv. Akad. Nauk SSSR, Ser. Mat., 28:307–364, 1964. W. Philipp, Arithmetic functions and Brownian motion, Proc. Symp. Pure Math., 24:233–246, 1973. A. Rényi, Probabilistic methods in number theory, in Proceedings of the International Congress of Mathematics, Edinburgh, August 14–21, 1958, Cambridge Univ. Press, Cambridge, 1960, pp. 529–539. I.Z. Ruzsa, Generalized moments of additive functions, J. Number Theory, 18(1):27–33, 1984. I.Z. Ruzsa, Generalizations of Kubilius’ class of additive functions. I, in F. Schweiger and E. Manstavičius (Eds.), Proceedings of the First International Conference in Honour of J. Kubilius, Palanga, Lithuania, 24–28 September 1991, New Trends Probab. Stat., VSP/TEV, Utrect/Vilnius, 1992, pp. 269–283. J. Šiaulys and V. Stakėnas, The Kubilius inequality for additive functions of rational argument, Liet. Mat. Rink., 30(1):176–184, 1990. V. Stakėnas and J. Šiaulys, Distribution of values of additive functions with rational argument, Liet. Mat. Rink., 28(3):565–581, 1988. G. Tenenbaum, Crible d’Eratosthène et modèle de Kubilius, in K. Györy, H. Iwaniec, and J. Urbanowicz (Eds.), Number Theory in Progress, Proceedings of the Conference in Honor of Andrzej Schinzel, Zakopane, Poland, June 30–July 9, 1997, Walter de Gruyter, Berlin, New York, 1999, pp. 1099–1129. P. Turán, On a theorem of Hardy and Ramanujan, J. Lond. Math. Soc., 9:274–276, 1934. P. Turán, On a theorem of Hardy and Ramanujan, J. Lond. Math. Soc., 11:125–133, 1936. R.V. Uždavinys, On the distribution of values of additive functions on the integer-valued polynomials, Proceedings of the Academy of Sciences of Lithuania, Ser. B, 2(18):9–29, 1959.