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Integral Limit Laws for Additive Functions

  • J. Galambos (a1)

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In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

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References

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1. Galambos, J., A probabilistic approach to mean values of multiplicative functions, J. London Math. Soc. 2 (1970), 405419.
2. Galambos, J., Distribution of arithmetical functions. A survey, Ann. Inst. H. Poincaré, Sect. B. 6 (1970), 281305.
3. Galambos, J., Distribution of additive and multiplicative functions, The theory of Arithmetic Functions, Lecture Notes Series (Springer Verlag, Vol. 251, 1972, pp. 127139).
4. Halàsz, G., Ûber die MittelwertemultiplikativerzahlentheoretischerFunktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365403.
5. Kubilius, J., Probabilistic methods in the theory of numbers, Transi. Math.Monographs, Amer. Math.Soc. 11, 1964.
6. Levin, B. V. and Fainleib, A. S., Applications of some integral equations to problems of number theory, Russian Math. Surveys 22 (1967), 119204.
7. Loéve, M., Probability theory, 3rd ed. (Van Nostrand, Princeton, N.J., 1963).
8. Titchmarsh, E. C., The theory of the Riemann zeta-function (Claredon Press, Oxford, 1951)..
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Integral Limit Laws for Additive Functions

  • J. Galambos (a1)

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