Distribution of arithmetical functions. A survey

 Access to full text

 Full (PDF)

1. Barban, M.B. [1] Arithmetical functions on « rare » sets (Russian), Trudy Insta. Mat., Akad. Nauk Uzbek. SSR, t. 22, 1961, p. 1-20. 
2. [2] The large sieve method and its application to number theory (Russian), Uspehi Mat. Nauk, t. 21, 1966, n° 1 (127), p. 51-102. Zbl0234.10031MR199171
3. Barban, M.B., Vinogradov, A.I. and Levin B.V. [1] Limit laws for functions of the class H of KUBILIUS J. P. which are defined on a set of « shifted » primes (Russian). Litovsk Mat. Sb., t. 5, 1965, p. 5-8. Zbl0146.05204MR205969
4. Behrend, F. [1] Über numeri abundantes, I. II, Sitzungsber. Preuss. Akad. Wiss. phys. math. Kl, 1932, 322-328 ; 1933, 280-293. Zbl59.0169.05JFM58.0158.03
5. Billingsley, P. [1] An application of Prohorov's theorem to probabilistic number theory, Illinois J. Math., t. 8, 1964, p. 697-704. Zbl0128.04302MR171769
6. [2] On the central limit theorem for the prime divisor function, Amer. Math. Monthly, t. 76, 1969, p. 132-139. Zbl0182.07401MR242222
7. Danilov, A.N. [1] On the sequences of values of additive arithmetical functions defined on the set of ideals of a field K of degree n over the field of rational numbers (Russian), Leningrad Gosudarst. Ped. Inst. A. I. Gercen, ucenye zapiski, t. 274, 1965, p. 273-304. MR197430
8. Davenport, H. [1] Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss, Phys-math., Kl., t. 27, 1933, p. 830-837. Zbl0008.19701
9. Delange, H. [1] Sur le nombre des diviseurs premiers de n, C. R. Acad. Sci., t. 237, 1953, p. 542-544. Zbl0051.03501MR57282
10. [2] Sur un théorème d'Erdôs et Kac, Acad. roy. Belg. Bull., El. Sci., t. 42, 1956, p. 130-144. Zbl0072.03501MR76800
11. [3] Sur les fonctions arithmétiques fortement additives, C. R. Acad. Sci., t. 244, 1957, p. 1307-1309; p. 1604-1606 ; p. 2122-2124. Zbl0077.05304
12. [4] Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Normale Supérieure, t. 78, 1961, p. 273-304. Zbl0234.10043MR169829
13. [5] Application de la méthode du crible à l'étude des valeurs moyennes de certaines fonctions arithmétiques, Séminaire Delange-Pisot (mimeographed lecture notes), Paris, 1962. 
14. [6] Acta Arith., t. 7, 1962, p. 191-215. 
15. [7] Sur un théorème de Rényi, Acta Arith., t. 11, 1965, p, 241-252. Zbl0131.04504MR201398
16. [8] Sur un théorème de Rényi II, Acta Arith., t, 13, 1968, p. 339-362. Zbl0155.09101MR228457
17. Delange, H. and Halberstam H. [1] A note on additive functions, Pacific J. Math., t. 7, 1957, p. 1551-1556. Zbl0080.26202MR92820
18. Elliott, P.D.T.A. [1] On certain additive functions, Acta Arith., t. 12, 1967, p. 365-384. [2] On certain additive functions II, Acta Arith., t. 14, 1968, p. 51-64. Zbl0147.29901MR211975
19. Erdös, P. [1] On the density of some sequences of numbers III, J. London Math. Soc., t. 13, 1938, p. 119-127. Zbl0018.29301
20. [2] On the smoothness of the asymptotic distribution of additive arithmetical functions, Amer. J. Math., t. 61, 1939, p. 722-725. Zbl0022.01001MR248JFM65.0165.02
21. Erdös, P. and Kac, M. [1] The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math., t. 62, 1940, p. 738-742. Zbl0024.10203MR2374
22. Erdös, P. and Wintner, A. [1] Additive arithmetical functions and statistical independence, Amer. J. Math., t. 61, 1939, p. 713-721. Zbl0022.00903MR247
23. Fainleib, A.S. [1] Distribution of values of Euler's function (Russian), Mat. Zametki, t. 1, 1967, p. 645-652. Zbl0209.35101MR215801
24. [2] On some asymptotic formulas for sums of multiplicative functions and applications (Russian), Lietuvos matematikos rinkinys, t. 7, 1967, p. 535-546. Zbl0209.35002MR228458
25. Galambos, J. [1] On the distribution of number theoretical functions, Annales Univ. Sci. Budapest, Sectio Math., t. 7, 1964, p. 25-31. Zbl0133.30001MR175868
26. [2] On the distribution of prime-independent additive number theoretical functions, Arch. Math., t. 19, 1968, p. 296-299. Zbl0159.05804MR228459
27. [3] Limit distribution of sums of (dependent) random variables with applications to arithmetical functions. To appear in Z. Wahrscheinlichkeitstheorie verw. Geb. Zbl0204.51206
28. [4] A probabilistic approach to mean values of multiplication functions, J. London Math. Soc. (2), t. 2, 1970, p. 405-419. Zbl0215.35802MR274411
29. Grigelionis, B. [1] The distribution of values of multiplicative arithmetical functions (Lithuanian), Vilniaus Valst-Univ. Mokslo Darbai Mat. Fiz., t. 9, 1960, p. 71-73. MR190122
30. [2] On the distribution of values of multiplicative functions (Russian), Litovsk Mat. Sb., t. 2, 1962, p. 107-125. Zbl0127.27403MR155811
31. Gyapjas, F. and Kátai, I. [1] Zur Statistik der Primfaktoren der natürlichen Zahlen, Annales Univ. Sci. Budapest, Sectio Math., t. 7, 1964, p. 59-66. Zbl0144.28402MR175876
32. Halász, G. [1] Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hung., t. 19, 1968, p. 365-402. Zbl0165.05804MR230694
33. Halberstam, H. [1] On the distribution of additive number theoretic functions I, J. London Math. Soc., t. 30, 1955, p. 43-53. Zbl0064.04202MR66406
34. Hardy, G.H. and Ramanujan, S. [1] The normal number of prime factors of a number n, Quart J. Pure and appl. Math., t. 48, 1917, p. 76-92. Zbl46.0262.03JFM46.0262.03
35. Hardy G.H. and Wright E.M. [1] An introduction to the theory of numbers, Oxford Univ. Press, 3rd ed., 1954. Zbl0058.03301MR67125
36. Iljasov, I. [1] The summation of composite functions of Euler's function (Russian), Dokl. Akad. Nauk SSSR, t. 178, 1968, p. 529-532. Zbl0172.05701MR224577
37. Juskis, Z. [1] Limit theorems for additive functions defined on ordered semigroups with a regular norm (Russian), Litovsk Mat. Sb., t. 4, 1964, p. 565-603. MR174540
38. [2] An asymptotic decomposition of distribution laws for certain functions defined on ordered semigroups with regular normalization (Russian), Litovsk Mat. Sb., t. 5, 1965, p. 167-183. Zbl0146.27203MR205970
39. Kac, M. [1] Statistical independence in probability, analysis and number theory, Monographs of Amer. Math. Soc., 1959. MR110114
40. Kátai, I. [1] Egy megjegyzés D. DELANGE : « Sur un théorème de Rényi » cimû dolgozatához (Hungarian), MTA III, Oszt. Mat. Fiz. , Közl, t. 16, 1966, p. 269-273. Zbl0145.04904MR237444
41. [2] On the distribution of arithmetical functions on the set of prime plus one, Compositio Math., 1969, to appear. Zbl0167.04301MR244181
42. [3] On the distribution of arithmetical functions, Acta Math. Acad. Sci. Hung., t. 20, 1969, p. 69-87. Zbl0175.04103MR237446
43. De Kroon, J.P.M. [1] The asymptotic behaviour of additive functions in algebraic number theory, Compositio Math., t. 17, 1965, p. 207-261. Zbl0143.06401MR205971
44. Kubilius, J. [1] On the distribution of values of additive arithmetical functions (Russian), Dokl. Akad. Nauk SSSR, t. 100, 1955, p. 623-626. Zbl0068.26703MR69417
45. [2] Probabilistic methods in number theory (Russian), Uspehi Mat. Nauk, t. 11, 1956, n° 2, p. 31-66. Zbl0071.04301MR79025
46. [3] On some problems in probabilistic number theory (Russian), Proc. Sixth All-Union Conf. Theory prob. and Math. Statist, Vilnius, 1960, p. 57-68. MR190123
47. [4] Probabilistic methods in number theory (Russian), and ed., Vilnius, 1962. 
48. [5] Probabilistic methods in the theory of numbers, Amer. Math. Soc., Translations of Math. monographs Vol. 11, 1964, p. 1-182. Zbl0133.30203MR160745
49. [6] On asymptotic laws of distribution of additive arithmetical functions (Russian), Litovsk Mat. Sb., t. 5, 1965, p. 261-273. MR204388
50. [7] On local theorems for additive number theoretic functions, Abhandlungen aus Zahlentheorie und analysis (Zur Erinnerung an E. Landau), Berlin, 1968, p. 175-191. Zbl0221.10058
51. Leveque, W.J. [1] On the size of certain number theoretic functions, Trans. Amer. Math. Soc., t. 66, 1949, p. 440-463. Zbl0036.30501MR30993
52. Levin, B.V. and Fainleib, A.S. [1] Distribution of values of additive arithmetical functions (Russian), Dokl. Akad. Nauk SSSR, t. 171, 1966, p. 281-284. Zbl0159.05901MR205972
53. [2] Application of some integral equations to questions of number theory (Russian), Uspehi Math. Nauk SSSR, t. 22 (135), 1967, p. 119-197. Zbl0204.06502MR229600
54. Misjavicjus, G. [1] The use of the method of moments in probabilistic number theory (Russian), Litousk Mat. Sb, t. 5, 1965, p. 275-289. Zbl0161.04505MR205973
55. Novoselov, E.V. [1] Topological theory of the divisors of integers (Russian), Uc. Zap. Elabuzsk. Pedag. Inst., t. 8, 1960, p. 3-23. 
56. [2] A new method in probabilistic number theory (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., t. 28, 1964, p. 307-364. MR168544
57. Paul, E.M. [1] Density in the light of probability theory I-III, Sankhya, Ser. A, t. 24, 1962, p. 103- 114, 209-212; t. 25, 1963, p. 273-280. Zbl0105.33303MR170357
58. Rényi, A. [1] On the density of certain sequences of integers, Publ. Inst. Math. Acad. Serbe Sci., t. 8, 1955, p. 157-162. Zbl0066.03203MR76787
59. [2] On the distribution of values of additive number theoretical functions, Publ. Math., Debrecen, t. 10, 1963, p. 264-273. Zbl0132.28504MR166136
60. [3] A new proof of a theorem of Delange, Publ. Math., Debrecen, t. 12, 1965, p. 323-329. Zbl0139.27003MR190124
61. [4] Additive and multiplicative number theoretic functions, Lecture Notes, University of Michigan (Ann. Arbor), 1965, p. 1-123. 
62. Rényi, A. and Turan, P. [1] On a theorem of Erdös-Kac, Acta Arith., t. 4, 1957, p. 71-84. Zbl0083.26401MR96629
63. Rieger, G.J. [1] Zur Statistik der Primfaktoren der natürlichen Zahlen in arithmetischen Progressionen, J. reine u. angew Math., t. 206, 1961, p. 26-30. Zbl0099.03004MR126434
64. [2] Über die Anzahl der Primfaketoren algebraischer Zahlen und das Gaussche Fehlergesetz, Math. Nachr., t. 24, 1962, p. 77-89. Zbl0286.12002MR144873
65. [3] Über die mittlere Anzahl der Primfaktoren von p-1, J. reine u. angew. Math., t. 210, 1962, p. 113-120. Zbl0273.10039MR148625
66. [4] Zum Teiler problem von Atle Selberg, Math. Nachr., t. 30, 1965, p. 181-192. Zbl0131.04603MR190105
67. Remond, P. [1] Évaluations asymptotiques dans certains semi-groupes, C. R. Acad. Sci., t. 260, 1965, p. 6250-6251. Zbl0253.10040MR177966
68. Robinson, R.L. [1] An estimate for the enumerative functions of certain sets of integers, Proc. Amer. Math. Soc., t. 17, 1966, p. 232-237. Zbl0144.28301MR184915
69. Sathe, L.G. [1] On a problem of Hardy on the distribution of integers having a given number of prime factors I-IV, J. Indian Math. Soc., t. 17, 1953, p. 63-82, p. 83-141 ; 18, 1954, p. 27-42, p. 43-81. Zbl0050.27102
70. Schönberg, I.J. [1] On asymptotic distribution of arithmetical functions, Trans. Amer. Math. Soc., t. 39, 1936, p. 315-330. Zbl0013.39302MR1501849
71. Selberg, A. [1] Note on a paper by L. G. Sathe, J. Indian Math. Soc., t. 18, 1954, p. 83-87. Zbl0057.28502MR67143
72. Shapiro, H.N. [1] Distribution functions of additive arithmetic functions, Proc. Nat. Acad. Sci. U. S. A., t. 42, 1956, p. 426-430. Zbl0071.04202MR79609
73. Tanaka, M. [1] On the number of prime factors of integers. I, IV, Japan. J. Math., t. 25, 1955, p. 1-20 ; J. Math. Soc. Japan., t. 9, 1957, p. 171-191 ; Japan. J. Math., t. 27, 1958, p. 103-127 ; t. 30, 1960, p. 55-83. Zbl0108.04503MR89225
74. Tjan, M.M. [1] On the question of the distribution of values of the Euler function p(n) (Russian), Litvosk Mat. Sb., t. 6, 1966, p. 105-119. MR205955
75. Turan, P. [1] On a theorem of Hardy and Ramanujan, J. London Math. Soc., t. 9, 1934, p. 274-276. Zbl0010.10401
76. [2] Über einige Verallgemeinerungen eines Satzes von Hardy and Ramanujan, J. London Math. Soc., t. 11, 1936, p. 125-133. Zbl0014.00901JFM62.0158.02
77. Uzdavinys, R. [1] On the distribution of values of additive arithmetical functions of polynomials (Russian), Trudy Akad. Nauk Litov. SSR, Ser. B, t. 18, 1959, p. 9-29. 
78. [2] Arithmetical functions on the set of values of integer valued polynomials (Russian), Litvosk Mat. Sb., t. 2, 1962, p. 253-280. MR157936
79. [3] An analogue theorem to that of Erdös and Wintner on the sequences of the values of integer valued polynomials (Russian), Litovsk Mat. Sb., t. 7, 1967, p. 329-338. MR228461
80. Vilkas, E. [1] On the normal distribution of additive arithmetical functions, Uc. Trudy viln. Gosud. Inst., t. 16, 1957, p. 23-31. 
81. Wirsing, E. [1] Das asymptotische Verhalten von Summen über multiplikative Funktionen II, Acta Math. Acad. Sci. Hung., t. 18, 1967, p. 411-467. Zbl0165.05901MR223318
82. Kubik, L. [1] The limiting distributions of cumulative sums of independent two-valued random variables, Studia Math., t. 18, 1959, p. 295-309. Zbl0113.34102MR119238
83. Lévy, P. [1] Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Math., t. 3, 1931, p. 119-155. Zbl0003.30301JFM57.0616.01
84. Loéve, M. [1] Probability theory, 3rd ed., Van Nostrand, Princeton, N. J., 1963 (lemmas 1, 3-6). Zbl0108.14202MR203748

1. János Galambos, On the asymptotic distribution of values of arithmetical functions