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Continuous Homomorphisms as Arithmetical Functions, and Sets of Uniqueness

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Part of the book series: Trends in Mathematics ((TM))

Abstract

This is a survey paper on the characterization of continuous group homomorphisms as arithmetical functions, and on sets of uniqueness with respect to completely additive functions.

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References

  1. P. Erdös, On the distribution function of additive functions, Ann. Math. 47 (1946), 1–20.

    Article  MATH  Google Scholar 

  2. I. Katai, A remark on additive arithmetical functions, Annales Univ. Sci. Budapest, Sectio Math. 10 (1967), 81–83.

    MathSciNet  MATH  Google Scholar 

  3. E. Wirsing, A characterization of log n as an additive arithmetic function, Symposia Mathematica, Instituto Nationale de Alta Mathematica I V (1970).

    Google Scholar 

  4. I. Katai, On a problem of P. Erdös, J. Number Theory 2 (1970), 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Wirsing, Characterization of the logarithm as an additive function, Proceedings of the 1969 Summer Institute of number theory, prepared by the Amer. Math. Soc. (1971), 375–381.

    Google Scholar 

  6. E. Wirsing, Additive and completely additive functions with restricted growth, Recent Progress in Analytic Number Theory, vol. 2, London, 1981, pp. 231–280.

    Google Scholar 

  7. I. Katai, On additive functions, Publ. Math. Debrecen 25 (1978), 251–257.

    MathSciNet  MATH  Google Scholar 

  8. I. Katai, Characterization of log n, Studies in Pure Mathematics (to the memory of Paul Turan), Akadémiai Kiadó, Budapest, 1984, pp. 415–421.

    Google Scholar 

  9. I. Katai, Some results and problems in the theory of additive functions, Acta Sci. Math. Szeged 30 (1969), 305–311.

    MathSciNet  MATH  Google Scholar 

  10. I. Katai, On number-theoretical functions, Colloquia Mathematica Sociates Janos Bolyai, vol. 2., North-Holland, Amsterdam, 1970, pp. 133–136.

    Google Scholar 

  11. J.L. Mauclaire, Sur la regularité des fonctions additives,Seminaire Delange-Pisot-Poitou, Theorie des Nombres, Paris 15 (1973/74), no. 23.

    Google Scholar 

  12. P.D.T.A. Elliott, On additive arithmetic function f (n) for which f (an + b)–f (cn + d) is bounded, J. Number Theory 16 (1983), 285–310.

    Article  MathSciNet  MATH  Google Scholar 

  13. P.D.T.A. Elliott, Arithmetic functions and integer products, Springer V., New York, 1985.

    Book  MATH  Google Scholar 

  14. A. Hildebrand, An Erdös-Wintner theorem for differences of additive functions, Trans. Amer. Math. Soc. 310 (1988), 257–276.

    MathSciNet  MATH  Google Scholar 

  15. P.D.T.A. Elliott, The value distribution of differences of additive arithmetic functions, J. Number Theory 32 (1989), 339–370.

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Katai and Indlekofer, K.-H., Estimation of generalized moments of additive functions over the set of shifted primes, Acta Sci. Math. 56 (1992), 229–236.

    MathSciNet  MATH  Google Scholar 

  17. P.D.T.A. Elliott, Sums and differences of additive arithmetic functions in mean square, J. reine und angewandte Mathematik 309 (1979), 21–54.

    MATH  Google Scholar 

  18. I. Katai, Multiplicative functions with regularity properties, I-VI, Acta Math. Hung. 42 (1983), 295–308; 43 (1984), 105–130, 259–272; 44 (1984), 125–132; 45 (1985), 379–380; 58 (1991), 343–350.

    Article  MathSciNet  Google Scholar 

  19. Katai, Arithmetical functions satisfying some relations, Acta Sci. Math. 55, 249–268.

    Google Scholar 

  20. I. Katai and Phong, B.M., On some pairs of mulitplicative functions correlated by an equation, New Trends in Probability and Statistics, Analytic and Probabilistic Methods in Number Theory, TEV, Vilnius, Lithuania 4, 191–203.

    Google Scholar 

  21. J. Fehér, Katai, I. and Phong, B.M., On multiplicative functions satisfying a special relation, Acta Sci. Math. (accepted).

    Google Scholar 

  22. I. Katai, Some problems in number theory, Studia Scient. Math. Hung. 16 (1981), 289–295.

    MathSciNet  MATH  Google Scholar 

  23. A. Hildebrand, Multiplicative functions at consecutive integers II., Math. proc. Cambridge Phil. Soc. 103 (1988), 389–398.

    Article  MathSciNet  MATH  Google Scholar 

  24. K.-H. Indlekofer and Katai, I., Multiplicative functions with small increments III., Acta Math. Hung. 58 (1991), 121–132.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. Wirsing, Tang Yuansheng and Shao Pintsung, On a conjecture of Katai for additive functions, J. Number Theory 56 (1996), 391–395.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. Katai, Additive functions with regularity properties, Acta Sci. Math. 44 (1982), 299–305.

    MathSciNet  MATH  Google Scholar 

  27. B.M. Phong, A characterization of some arithmetical multiplicative functions, Acta Math. Hung. 63 (1994), 29–43.

    Article  MathSciNet  MATH  Google Scholar 

  28. N.L. Bassily and KStai, I., On the pairs of multiplicative functions satisfying some relations, Aequationes Math. 55 (1998), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  29. Tang Yuansheng, A reverse problem on arithmetic functions, J. Number Theory 58 (1996), 130–138.

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Kâtai, On additive functions satisfying a congruence, Acta Sci. Math. 47 (1984), 85–92.

    MATH  Google Scholar 

  31. R. Styer, A problem of Kâtai on sums of additive functions, Acta Sci. Math. 55 (1991), 269–286.

    MathSciNet  MATH  Google Scholar 

  32. M. Wijsmuller, Additive functions on the Gaussian integers, Publ. Math. Debrecen 38 (1991), 255–262.

    MathSciNet  MATH  Google Scholar 

  33. I. Kâtai and Wijsmuller, M., Additive functions satisfying congruences, Acta Sci. Math. 56 (1992), 63–72.

    MATH  Google Scholar 

  34. I.Kâtai and Subbarao, M.V., The characterization of n ir as a multiplicative function, Acta Math. Hung. (accepted).

    Google Scholar 

  35. I.Kâtai and Subbarao, M.V., On the multiplicative function nil, Studia Sci. Math. (accepted).

    Google Scholar 

  36. Z. Daróczy and Katai, I., On additive arithmetical functions with values in the circle group, Publ. Math. Debrecen 34 (1987), 307–312.

    MathSciNet  MATH  Google Scholar 

  37. Z. Daróczy and Kâtai, I., On additive arithmetical functions with values in topological groups, Publ. Math. Debrecen 33 (1986), 287–292.

    MathSciNet  Google Scholar 

  38. Z. Daróczy and Kâtai, I., On additive number-theoretical functions with values in a compact Abelian group, Aequationes Math. 28 (1985), 288–292.

    Article  MathSciNet  MATH  Google Scholar 

  39. Z. Daróczy and Kâtai, I., On additive arithmetical functions with values in topological groups II, Publ. Math. Debrecen 34 (1987), 65–68.

    MathSciNet  Google Scholar 

  40. Z. Daróczy and Kâtai, I., On additive functions taking values from a compact group, Acta Sci. Math. 53 (1989), 59–65.

    MATH  Google Scholar 

  41. Z. Daróczy and Katai, I., Characterization of additive functions with values in the circle group, Publ. Math. Debrecen 36 (1989), 1–7.

    MathSciNet  MATH  Google Scholar 

  42. B.M. Phong, Note on multiplicative functions with regularity properties, Publ. Math. Debrecen 41 (1992), 117–125.

    MathSciNet  MATH  Google Scholar 

  43. B.M. Phong, Characterization of additive functions with values in a compact Abelian group, Publ. Math. Debrecen 40 (1992), 273–278.

    MathSciNet  MATH  Google Scholar 

  44. Z. Daróczy and Kâtai, I., Characterization of pairs of additive functions with some regularity property, Publ. Math. Debrecen 37 (1990), 217–221.

    MathSciNet  MATH  Google Scholar 

  45. J.L. Mauclaire, On the regularity of group valued additive arithmetical functions, Publ. Math. Debrecen 44 (1994), 285–290.

    MathSciNet  MATH  Google Scholar 

  46. I.Z. Ruzsa and Tijdeman, R., On the difference of integer-valued additive functions, Publ. Math. Debrecen 39 (1991), 353–358.

    MathSciNet  MATH  Google Scholar 

  47. I. Kdtai, On sets characterizing number-theoretical functions, Acta Arithm. 13 (1968), 315–320.

    Google Scholar 

  48. I. Kâtai, On sets characterizing number-theoretical functions II (The set of “prime plus one”s is a set of quasi uniqueness), Acta Arithm. 16 (1968), 1–4.

    Google Scholar 

  49. P.D.T.A. Elliott, A conjecture of Kâtai, Acta Arithm. 26 (1974), 11–20.

    MATH  Google Scholar 

  50. D. Wolke, Bemerkungen über Eindeutigkeitsmengen additiver Functionen, Elem. der Math. 33 (1978), 14–16.

    MathSciNet  MATH  Google Scholar 

  51. F. Dress and Volkman, B., Ensembles d’unicité pour les fonctions arithemtiques additives ou multiplicatives, C. R. Acad. Sci. Paris Ser. A 287 (1978), 43–46.

    MathSciNet  MATH  Google Scholar 

  52. P.D.T.A. Elliott, On two conjectures of Katai, Acta Arithm. 30 (1976), 35–39.

    Google Scholar 

  53. E. Wirsing, Additive functions with restricted growth on the numbers of the form p -F 1, Acta Arithm. 37 (1980), 345–357.

    MathSciNet  MATH  Google Scholar 

  54. I. Kátai, On distribution of arithmetical functions on the set of prime plus one, Compositio Math. 19 (1968), 278–289.

    MathSciNet  MATH  Google Scholar 

  55. P.D.T.A. Elliott, On the limiting distribution of f (p + 1) for non-negative additive functions, Acta Arithm. 25 (1974), 259–264.

    MATH  Google Scholar 

  56. I. Kátai, Some remarks on additive arithmetical functions, Litovsk. Mat. Sb. 9 (1969), 515–518.

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Hildebrand, Additive and multiplicative functions on shifted primes Proc. London Math. Soc. 53 (1989), 209–232.

    Google Scholar 

  58. J. Meyer, Ensembles d’unicité pour les fonctions additives. Étude analogue dans les cas des fonctions multiplicatives,Journées de Théorie Analytique et Élémentaire des Nombres, Orsay, 2 et 3 Juin, 1980. Publications Mathématiques d’Orsay, 50–66.

    Google Scholar 

  59. K.-H. Indlekofer, On sets characterizing additive and multiplicative arithmetical functions, Illinois.1. of Math. 25 (1981), 251–257.

    MathSciNet  MATH  Google Scholar 

  60. P.D.T.A. Elliott, On representing integers as products of integers of a prescribed type, J. Australian Math. Soc. (Series A) 35 (1983), 143–161.

    Article  MATH  Google Scholar 

  61. P.D.T.A. Elliott, On representing integers as products of the p + 1, Monatschrifte fir Math. 97 (1984), 85–97.

    Article  MATH  Google Scholar 

  62. P.D.T.A. Elliott,The multiplicative group of rationale generated by the shifted primes, I (manuscript).

    Google Scholar 

  63. J. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176.

    MathSciNet  MATH  Google Scholar 

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Kátai, I. (2000). Continuous Homomorphisms as Arithmetical Functions, and Sets of Uniqueness. In: Bambah, R.P., Dumir, V.C., Hans-Gill, R.J. (eds) Number Theory. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7023-8_11

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  • DOI: https://doi.org/10.1007/978-3-0348-7023-8_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7025-2

  • Online ISBN: 978-3-0348-7023-8

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