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Integral Limit Laws for Additive Functions

Published online by Cambridge University Press:  20 November 2018

J. Galambos
J. Galambos*
Affiliation:
Temple University, Philadelphia, Pennsylvania
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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].

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 194 - 203
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Galambos, J., A probabilistic approach to mean values of multiplicative functions, J. London Math. Soc. 2 (1970), 405–419.Google Scholar
2. Galambos, J., Distribution of arithmetical functions. A survey, Ann. Inst. H. Poincaré, Sect. B. 6 (1970), 281–305.Google Scholar
3. Galambos, J., Distribution of additive and multiplicative functions, The theory of Arithmetic Functions, Lecture Notes Series (Springer Verlag, Vol. 251, 1972, pp. 127–139).Google Scholar
4. Halàsz, G., Ûber die MittelwertemultiplikativerzahlentheoretischerFunktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403.Google Scholar
5. Kubilius, J., Probabilistic methods in the theory of numbers, Transi. Math.Monographs, Amer. Math.Soc. 11, 1964.Google Scholar
6. Levin, B. V. and Fainleib, A. S., Applications of some integral equations to problems of number theory, Russian Math. Surveys 22 (1967), 119–204.Google Scholar
7. Loéve, M., Probability theory, 3rd ed. (Van Nostrand, Princeton, N.J., 1963).Google Scholar
8. Titchmarsh, E. C., The theory of the Riemann zeta-function (Claredon Press, Oxford, 1951)..Google Scholar