Probability methods in some problems of analysis and number theory

Kac, M.

doi:10.1090/S0002-9904-1949-09242-X

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-9485(online) ISSN 0273-0979(print)

Probability methods in some problems of analysis and number theory

Author: M. Kac
Journal: Bull. Amer. Math. Soc. 55 (1949), 641-665
MathSciNet review: 0031504
Full-text PDF

References | Additional Information

S. Banach 1. Ueber einige Eigenschaften der lakunären trigonometrischen Reihen, Studia Mathematica vol. 2 (1930) pp. 207-220.
H. Davenport 1. Ueber Numeri Abundantes, Preuss. Akad. Wiss. Sitzungsber. (1930) pp. 830-837.
P. Erdös, On the strong law of large numbers, Trans. Amer. Math. Soc. 67 (1949), 51–56. MR 0032971 (11,375c), http://dx.doi.org/10.1090/S0002-9947-1949-0032971-4
P. Erdös 2. Note on the number of prime divisors of integers, J. London Math. Soc. vol. 12 (1937) pp. 308-314.
P. Erdös, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1–20. MR 0015424 (7,416c)
P. Erdös 4. On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. vol. 32 (1936) pp. 530-540.
P. Erdös 5. On the density of some sequences of numbersI, J. London Math. Soc. vol. 10 (1935) pp. 120-125.
P. Erdös 6. On the density of some sequences of numbersII, ibid. vol. 12 (1937) pp. 7-11.
P. Erdös 7. On the density of some sequences of numbersIII, ibid. vol. 13 (1938) pp. 119-127.
P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742. MR 0002374 (2,42c)
Paul Erdös and Aurel Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713–721. MR 0000247 (1,40c)
W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc. 51 (1945), 800–832. MR 0013252 (7,128i), http://dx.doi.org/10.1090/S0002-9904-1945-08448-1
Jacqueline Ferrand and Robert Fortet, Sur des suites arithmétiques équiréparties, C. R. Acad. Sci. Paris 224 (1947), 516–518 (French). MR 0019258 (8,391f)
R. Fortet, Sur une suite egalement répartie, Studia Math. 9 (1940), 54–70 (French, with Ukrainian summary). MR 0005546 (3,169e)
G. H. Hardy and S. Ramanujan 1. The normal number of prime factors of a number n, Quart. J. Math. Oxford Ser. vol. 48 (1917) pp. 76-92.
Philip Hartman, The divergence of non-harmonic gap series, Duke Math. J. 9 (1942), 404–405. MR 0006768 (4,39b)
Shin-ichi Izumi and Tatsuo Kawata, On certain series of functions, Tôhoku Math. J. 46 (1939), 91–105. MR 0001322 (1,217f)
M. Kac, Note on Power Series with Big Gaps, Amer. J. Math. 61 (1939), no. 2, 473–476. MR 1507389, http://dx.doi.org/10.2307/2371514
M. Kac 2. Distribution properties of certain gap sequences, Bull. Amer. Math. Soc. Abstract 53-7-290. Details will be given in the paper by Erdös, Ferrand, Fortet and Kac (see footnote 2). The following misprint should be corrected: ω/2cos πk| should be replaced by ω/2|cos πx|.
M. Kac, On the distribution of values of sums of the type ∑𝑓(2^{𝑘}𝑡), Ann. of Math. (2) 47 (1946), 33–49. MR 0015548 (7,436f)
M. Kac, Convergence of certain gap series, Ann. of Math. (2) 44 (1943), 411–415. MR 0008432 (5,4c)
M. Kac, Convergence and divergence of non-harmonic gap series, Duke Math. J. 8 (1941), 541–545. MR 0005143 (3,107g)
M. Kac 6. Sur les fonctions indépendantesV, Studia Mathematica vol. 7 (1938) pp. 96-100.
M. Kac, Note on the distribution of values of the arithmetic function 𝑑(𝑚), Bull. Amer. Math. Soc. 47 (1941), 815–817. MR 0004845 (3,69e), http://dx.doi.org/10.1090/S0002-9904-1941-07575-0
M. Kac, R. Salem, and A. Zygmund, A gap theorem, Trans. Amer. Math. Soc. 63 (1948), 235–243. MR 0023937 (9,426a), http://dx.doi.org/10.1090/S0002-9947-1948-0023937-8
S. Kaczmarz and H. Steinhaus 1. Le système orthogonal de M. Rademacher, Studia Mathematica vol. 2 (1930) pp. 231-247.
A. Khintchine and A. Kolmogoroff 1. Ueber Konvergenz von Reihen deren Glieder durch den Zufall bestimmt werden, Rec. Math. (Mat. Sbornik) vol. 32 (1925) pp. 668-677.
A. Kolmogoroff, Über die Summen durch den Zufall bestimmter unabhängiger Größen, Math. Ann. 99 (1928), no. 1, 309–319 (German). MR 1512451, http://dx.doi.org/10.1007/BF01459098
A. Kolmogoroff 2. Bemerkungen zu meiner Arbeit "Ueber die summen Zufälliger Grössen", ibid. vol. 102 (1929) pp. 484-489.
A. Kolmogoroff 3. Une contribution à l'étude de la convergence des séries de Fourier, Fund. Math. vol. 5 (1924) pp. 96-97.
P. Lévy 1. Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Mathematica vol. 3 (1931) pp. 119-155.
W. J. LeVeque 1. On the distribution of values of number-theoretic functions, Thesis, Cornell University.
R. E. A. C. Paley and A. Zygmund 1. On some series of functions (1), Proc. Cambridge Philos. Soc. vol. 26 (1930) pp. 337-357.
R. E. A. C. Paley and A. Zygmund 2. On some series of functions (2), ibid. pp. 458-474.
R. E. A. C. Paley and A. Zygmund 3. On some series of functions (3), ibid. vol. 28 (1932) pp. 190-205.
Hans Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), no. 1-2, 112–138 (German). MR 1512104, http://dx.doi.org/10.1007/BF01458040
D. Raikov 1. On some arithmetical properties of summable functions, Rec. Math. (Mat. Sbornik) N.S. vol. 1 (1936) pp. 377-383.
Frédéric Riesz, Sur la théorie ergodique, Comment. Math. Helv. 17 (1945), 221–239 (French). MR 0014218 (7,255b)
R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 333–338. MR 0022263 (9,181d)
R. Salem and A. Zygmund, On lacunary trigonometric series. II, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 54–62. MR 0023936 (9,425h)
I. J. Schoenberg 1. Ueber die asymptötische Verteilung reeller Zahlenmod 1, Math. Zeit. vol. 28 (1928) pp. 171-200.
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Trans. Amer. Math. Soc. 39 (1936), no. 2, 315–330. MR 1501849, http://dx.doi.org/10.1090/S0002-9947-1936-1501849-X
H. Steinhaus 1. Les probabilités dénombrables et leur rapport à la théorie de la mesure, Fund. Math. vol. 4 (1922) pp. 286-310.
P. Turán 1. Ueber einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan, J. London Math. Soc. vol. 11 (1936) pp. 125-133.
A. Zygmund 1. On the convergence of lacunary trigonometric series, Fund. Math. vol. 16 (1930) pp. 90-107.

Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9904-1949-09242-X
PII: S 0002-9904(1949)09242-X

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|