Functional analysis and additive arithmetic functions
Author:
P. D. T. A. Elliott
Journal:
Bull. Amer. Math. Soc. 16 (1987), 179-223
MSC (1985):
Primary 11K65, 11N37, 47B99
DOI:
https://doi.org/10.1090/S0273-0979-1987-15507-8
MathSciNet review:
876958
Full-text PDF Free Access
References | Similar Articles | Additional Information
- E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 197425, DOI https://doi.org/10.1112/S0025579300005313
- Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
- François Dress and Bodo Volkmann, Ensembles d’unicité pour les fonctions arithmétiques additives ou multiplicatives, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 2, A43–A46 (French, with English summary). MR 491439
- P. D. T. A. Elliott, On inequalities of large sieve type, Acta Arith. 18 (1971), 405–422. MR 286774, DOI https://doi.org/10.4064/aa-18-1-405-422
- P. D. T. A. Elliott, On connections between the Turán-Kubilius inequality and the large sieve: some applications, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 77–82. MR 0344215
- P. D. T. A. Elliott, On the limiting distribution of additive arithmetic functions, Acta Math. 132 (1974), 53–75. MR 325562, DOI https://doi.org/10.1007/BF02392108
- P. D. T. A. Elliott, The law of large numbers for additive arithmetic functions, Math. Proc. Cambridge Philos. Soc. 78 (1975), 33–71. MR 387229, DOI https://doi.org/10.1017/S0305004100051495
- P. D. T. A. Elliott, General asymptotic distributions for additive arithmetic functions, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 43–54. MR 392892, DOI https://doi.org/10.1017/S0305004100052087
- P. D. T. A. Elliott, The Turán-Kubilius inequality, Proc. Amer. Math. Soc. 65 (1977), no. 1, 8–10. MR 447163, DOI https://doi.org/10.1090/S0002-9939-1977-0447163-7
- P. D. T. A. Elliott, On the differences of additive arithmetic functions, Mathematika 24 (1977), no. 2, 153–165. MR 472734, DOI https://doi.org/10.1112/S0025579300009049
- P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
- P. D. T. A. Elliott, High-power analogues of the Turán-Kubilius inequality, and an application to number theory, Canadian J. Math. 32 (1980), no. 4, 893–907. MR 590654, DOI https://doi.org/10.4153/CJM-1980-068-0
- P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 272, Springer-Verlag, New York, 1985. MR 766558
- P. D. T. A. Elliott, Cauchy’s functional equation in the mean, Adv. in Math. 51 (1984), no. 3, 253–257. MR 740585, DOI https://doi.org/10.1016/0001-8708%2884%2990010-0
- P. D. T. A. Elliott, The exponential function characterized by an approximate functional equation, Illinois J. Math. 26 (1982), no. 3, 503–518. MR 658460
- P. D. T. A. Elliott and C. Ryavec, The distribution of the values of additive arithmetical functions, Acta Math. 126 (1971), 143–164. MR 277491, DOI https://doi.org/10.1007/BF02392029
- P. Erdös, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1–20. MR 15424, DOI https://doi.org/10.2307/1969031 18. P. Erdős and M. Kac, On the Gaussian law of errors in the theory of additive functions, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 206-207.
- 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 2374, DOI https://doi.org/10.2307/2371483
- G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 230694, DOI https://doi.org/10.1007/BF01894515 21. G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. (Oxford) 48 (1917), 76-92.
- Adolf Hildebrand, Sur les moments d’une fonction additive, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 1–22 (French, with English summary). MR 723945
- Adolf Hildebrand, An asymptotic formula for the variance of an additive function, Math. Z. 183 (1983), no. 2, 145–170. MR 704101, DOI https://doi.org/10.1007/BF01214818
- I. Kátai, On a problem of P. Erdős, J. Number Theory 2 (1970), 1–6. MR 250991, DOI https://doi.org/10.1016/0022-314X%2870%2990002-8
- I. P. Kubilyus, Probabilistic methods in the theory of numbers, Amer. Math. Soc. Transl. (2) 19 (1962), 47–85. MR 0131415, DOI https://doi.org/10.1090/trans2/019/02
- J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745
- J. Kubilius, On an inequality for additive arithmetic functions, Acta Arith. 27 (1975), 371–383. MR 366859, DOI https://doi.org/10.4064/aa-27-1-371-383
- Ĭ. Kubilyus, Estimation of the second central moment for strongly additive arithmetic functions, Litovsk. Mat. Sb. 23 (1983), no. 1, 122–133 (Russian, with English and Lithuanian summaries). MR 705734
- Ĭ. Kubilyus, Estimate of the second central moment for any additive arithmetic functions, Litovsk. Mat. Sb. 23 (1983), no. 2, 110–117 (Russian, with English and Lithuanian summaries). MR 706013
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- B. V. Levin and N. M. Timofeev, An analytic method in probabilistic number theory, Vladimir. Gos. Ped. Inst. Učen. Zap. 38 (1971), 57–150 (Russian). MR 0302596
- Jacques Meyer, Ensembles d’unicité pour les fonctions additives. Étude analogue dans le cas des fonctions multiplicatives, Conference on Analytic and Elementary Number Theory (Orsay, 1980) Publ. Math. Orsay 81, vol. 1, Univ. Paris XI, Orsay, 1981, pp. 50–66 (French). MR 610263
- Karl Prachar, Primzahlverteilung, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 91, Springer-Verlag, Berlin-New York, 1978 (German). Reprint of the 1957 original. MR 516660
- Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725
- I. Z. Ruzsa, General multiplicative functions, Acta Arith. 32 (1977), no. 4, 313–347. MR 432572, DOI https://doi.org/10.4064/aa-32-4-313-347
- I. Z. Ruzsa, On the concentration of additive functions, Acta Math. Acad. Sci. Hungar. 36 (1980), no. 3-4, 215–232 (1981). MR 612193, DOI https://doi.org/10.1007/BF01898136 37. I. Z. Ruzsa, On the variance of additive functions, Turán Memorial Volume of the J. Bolyai Math. Soc., Hungary.
- Imre Z. Ruzsa, Generalized moments of additive functions, J. Number Theory 18 (1984), no. 1, 27–33. MR 734434, DOI https://doi.org/10.1016/0022-314X%2884%2990039-8 39. P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274-276. 40. P. Turán, Über einige Verallgemeinerungen eines Satzes von Hardy and Ramanujan, J. London Math. Soc. 11 (1936), 125-133.
- E. Wirsing, A characterization of ${\rm l0g}$ $n$ as an additive arithmetic function, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 45–57. MR 0271049
- Eduard Wirsing, Characterisation of the logarithm as an additive function, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N. Y., 1969) Amer. Math. Soc., Providence, R. I., 1971, pp. 375–381. MR 0412129
- Dieter Wolke, Bemerkungen über Eindeutigkeitsmengen additiver Funktionen, Elem. Math. 33 (1978), no. 1, 14–16 (German). MR 463096 44. K. Yosida, Functional analysis, Grundlehren math. Wiss., vol. 123, 3rd éd., Springer-Verlag, New York, Berlin, Heidelberg, 1971.
Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 11K65, 11N37, 47B99
Retrieve articles in all journals with MSC (1985): 11K65, 11N37, 47B99
