Limit theorems for divisor distributions. (English) Zbl 0609.10041

Vose, Michael D.

Proc. Am. Math. Soc. 95, 505-511 (1985).

Let N be a positive integer and for \(N=p_ 1^{\alpha_ 1}... p_ k^{\alpha_ k}\) define \[ \mu_ n=(\sum_{1\leq j\leq k}((\alpha_ j+1)^ n-1)(\log p_ j)^ n)^{1/n},\quad F_ N(x)=\sum_{\alpha | N}'1/\tau (N), \] where \(\sum '\) denotes that the sum is restricted by those divisors satisfying log(d/\(\sqrt{N})\leq x\mu_ 2\). In this paper necessary and sufficient criteria for the convergence of \(F_{N_ j}\), \(N_ j=\prod_{p\leq j}p\), are given as \(j\to \infty\). In the general case it is proved that the necessary and sufficient conditions for a sequence \(F_{N_ j}\) to converge to the distribution F is that for each n the limits \(\lim_{j\to \infty}\mu_{2n}(N_ j)(\mu_ 2(N_ j))^{-1}\) exist. The possible limit distributions, that arise in this case, are investigated.
If \(F(x)=\phi (x)\)- normal distribution - then \(\sup_{w}| F_ N(w)-\phi (w)| \leq \mu_{\infty}/\mu_ 2\), where \(\mu_{\infty}=\lim_{n\to \infty}\mu_ n\).

Reviewer: B.V.Levin

MSC:

11K65	Arithmetic functions in probabilistic number theory
11N37	Asymptotic results on arithmetic functions

Keywords:

limit theorems; divisor distributions; divisors; limit distributions

References:

[1]	John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. · Zbl 0277.30001
[2]	Paul Erdős and Jean-Louis Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Bull. Sci. Math. (2) 100 (1976), no. 4, 301 – 320 (French). · Zbl 0343.10037
[3]	William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0077.12201
[4]	D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. · Zbl 0199.38101