Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 409.10039
Autor: Elliott, P.D.T.A.; Erdös, Paul
Title: The tails of infinitely divisible laws and a problem in number theory. (In English)
Source: J. Number Theory 11, 542-551 (1979).
Review: A function defined on the positive integers is said to be additive if it satisfies the relation f(mn) = f(m)+f(n) for all relatively prime m and n. An example of an additive function is \omega(n), the number of distinct prime divisors of the positive integer n. A great deal of work has gone into proving central limit type theorem for additive functions satisfying certain moment conditions. See P.Erdös and M.Kac [Am. J. Math. 62, 738-742 (1940; Zbl 024.10203)] for early such work, and {\it J.Kubilius's monograph [Probabilistic methods in the theory of numbers (1962; Zbl 127.27402)]. See also W.Philipp [Analytic Number Theory, Proc. Symp. pure Math. 24, St. Louis Univ. Missouri 1972, 233-246 (1973; Zbl 269.10031)] for functional central limit theorems for additive functions. In many of these investigations the theory of sums of independent random variables come into play, and the limit laws are infinitely divisible.
In the present paper the additive function f generated by f(pm) = (log pm)\alpha with 0 < \alpha < 1 is considered, and the limit law. K\alpha8x) = (weak)limN ––> ooN-1\sharp{n \leq N| f(n) < x(log N)\alpha} (*) is studied. P. Erdös in [Ann. of Math., II Ser. 47, 1-20 (1946; Zbl 061.07902) ] had noted that K\alpha exists for all \alpha. If \alpha = 1, then K\alpha is degenerate (consists of a single jump). For \alpha > 1, B.V.Levin and N.M.Timofeev [Acta arithmetica 26, 333-364 (1975; Zbl 318.10041)] have shown that the support of K\alpha lies in the unit interval. Non-degenerate laws with bounded support are not infinitely dibisible, and hence K\alpha is not infinitely divisible in this case. For 0 < \alpha < 1 it turns out the support of K\alpha is not concentrated on a bounded interval but the authors show that K\alpha is nevertheless not infinitely divisible. The proof depends on a probabilistic theorem of independent interest. The authors show that if the tail of an infinitely divisible probality law approaches zero sufficiently rapidly, then the law must be the normal law. For K\alpha to be infinitely divisible, it must be normal, which it clearly is not. Hence K\alpha is not infinitely divisible for any positive \alpha\ne 1. This implies that any attempt to study K\alpha by applying the theory of sums of independent random variables to (*) in a straightforward way seems doomed to failure. For a general picture of probabilistic methods in the theory of additive functions see P.Billingsley [Ann. of Probab. 2, 749-791 (1974; Zbl 327.10055)].
Reviewer: O.P.Stackelberg
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
60F05 Weak limit theorems
Keywords: infinitely divisible probability law; normal law; distribution function; arithmetic functions; additive functions; limit laws
Citations: Zbl.327.10055; Zbl.024.102; Zbl.127.274; Zbl.263.10019; Zbl.061.079; Zbl.318.10041
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