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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.
- 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 1. On the strong law of large numbers, Bull. Amer. Math. Soc. Abstract 54-5-191. The paper will appear in Trans. Amer. Math. Soc. MR 32971
- P. Erdös 2. Note on the number of prime divisors of integers, J. London Math. Soc. vol. 12 (1937) pp. 308-314.
- P. Erdös 3. On the distribution function of additive functions, Ann. of Math. vol. 47 (1946) pp. 1-20. MR 15424
- 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 1. The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. vol. 62 (1940) pp. 738-742. MR 2374
- P. Erdös and A. Wintner 1. Additive arithmetical functions and statistical independence, Amer. J. Math. vol. 61 (1939) pp. 713-721. MR 247
- W. Feller 1. The fundamental limit theorems in probability, Bull. Amer. Math. Soc. vol. 51 (1945) pp. 800-832. MR 13252
- J. Ferrand and R. Fortet 1. Sur des suites arithmétiques equiréparties, C. R. Acad. Sci. Paris vol. 224 (1947) pp. 514-515. MR 19258
- R. Fortet 1. Sur une suite également repartie, Studia Mathematica 9 (1940) pp. 54-69. MR 5546
- 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.
- P. Hartman 1. The divergence of non-harmonic gap series, Duke Math. J. vol. 9 (1942) pp. 404-405. MR 6768
- S. Izumi and T. Kawata 1. On certain series of functions, Tôhoku Math. J. vol. 46 (1939) pp. 91-105. MR 1322
- M. Kac 1. Note on power series with big gaps, Amer. J. Math. vol. 61 (1939) pp. 473-476. MR 1507389
- 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 3. On the distribution of values of sums of the typeΣf(2, Ann. of Math. vol. 47 (1946) pp. 33-49. MR 15548
- M. Kac 4. Convergence of certain gap series, ibid. 44 (1943) pp. 411-415. MR 8432
- M. Kac 5. Convergence and divergence of non harmonic gap series, Duke Math. J. vol. 8 (1941) pp. 541-545. MR 5143
- M. Kac 6. Sur les fonctions indépendantesV, Studia Mathematica vol. 7 (1938) pp. 96-100.
- M. Kac 7. Note on the distribution of values of the arithmetic function d(m), Bull. Amer. Math. Soc. vol. 47 (1941) pp. 815-817. MR 4845
- M. Kac, R. Salem and A. Zygmund 1. A gap theorem, Trans. Amer. Math. Soc. vol. 63 (1948) pp. 235-243. MR 23937
- 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 1. Ueber die Summen durch den Zufall bestimmter unabhängiger Grössen, Math. Ann. vol. 99 (1928) pp. 309-319. MR 1512451
- 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.
- H. Rademacher 1. Einige Sätze ueber Reihen von allgemeinenen Orthogonalfunktionen, Math. Ann. vol. 87 (1922) pp. 112-138. MR 1512104
- D. Raikov 1. On some arithmetical properties of summable functions, Rec. Math. (Mat. Sbornik) N.S. vol. 1 (1936) pp. 377-383.
- F. Riesz 1. Sur la théorie ergodique, Comment. Math. Helv. vol. 17 (1945) pp. 221-239. MR 14218
- R. Salem and A. Zygmund 1. On lacunary trigonometric series, Proc. Nat. Acad. Sci. U.S.A. vol. 33 (1947) pp. 333-338. MR 22263
- R. Salem and A. Zygmund 2. On lacunary trigonometric series, ibid. vol. 34 (1948) pp. 54-62. MR 23936
- I. J. Schoenberg 1. Ueber die asymptötische Verteilung reeller Zahlenmod 1, Math. Zeit. vol. 28 (1928) pp. 171-200.
- I. J. Schoenberg 2. On asymptotic distribution of arithmetical functions, Trans. Amer. Math. Soc. vol. 39 (1936) pp. 315-330. MR 1501849
- 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
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