Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 431.10031
Autor: Erdös, Paul; Katai, I.
Title: On the concentration of distribution of additive functions. (In English)
Source: Acta Sci. Math. 41, 295-305 (1979).
Review: The authors investigate the singularity or absolute continuity of distribution functions for certain classes of additive functions. A number-theoretic function f is said to be additive if f(mn) = f(m)+f(n) whenever m and n are relatively prime positive integers. Also, f is strongly additive if f(pk) = (f(p))k for all primes p. The function f has a limiting distribution if the frequencies (1/n)\sharp{n| f(n) \leq x} converge weakly to a distribution function F(x). Necessary and sufficient conditions for the existence and continuity of distribution functions of additive functions are classical by now, but it seems to be a difficult problem to distinguish between absolute continuity and singularity in particular cases. For an interesting discussion of these problems refer to the book by P.D.T.A.Elliott [Probabilistic number theory. I, II (1979 und 1980; Zbl 431.10029 und 431.10030)]. The main results of the paper are contained in four theorems. If F is distribution function, then Q(h) = \supx(F(x+h)-F(x)) is called the concentration function of F. The authors prove some results involving concentration functions of a class of strongly additive functions, thereby generalizing corresponding theorems for some particular additive functions. We state one of these results here. Theorem: Let D(y) = sump < y |f(p)|/p, and suppose that D(tc) < 1/t and |f(p1)-f(p29| > 1/t if p1\ne p2 < t\delta hold, for suitable positive constants c and \delta, for every large t. Then (log t)-1 << Q(log t)-1 as t ––> oo. This theorem was proved for f(n) = log(\phi(n)/n) by M.M.Tjan [Litov. Mat. Sb. 6, 105-119 (1966; Zbl 163.29201)] , and for log(\sigma(n)/n) by P.Erdös [Pac. J. Math. 52, 59-65 (1974; Zbl 291.10040)].
Reviewer: O.P.Stackelberg
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
Keywords: singularity; absolute continuity; distribution functions; additive functions; concentration function
Citations: Zbl.431.10029; Zbl.431.10030; Zbl.163.292; Zbl.291.10040
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