The distribution of divisors of \(N!\). (English) Zbl 0647.10038

Vose, Michael D.

Acta Arith. 50, No. 2, 203-209 (1988).

The author is interested in probabilistic aspects of the distribution of the divisors of \((N!)\). Starting with random variables \(X_j\), uniformly distributed over the set \(\{\log d, d/j!\},\) with distribution functions \(F_j\), the author shows: The sequence \(F_j\) converges to the distribution \(\psi\) with density \(\rho\), represented by the infinite convolution \(\rho =Y_1*Y_2*\cdots\), where \(Y_i=(2\xi_i)^{- 1}\chi_{[-\xi_i,\xi_i]},\xi_i=\frac{\sigma \log p_i}{p_i-1},\sigma =(\frac{1}{3}\sum_{p}(p-1)^{-2}\cdot \log^2p)^{-1/2}.\) Moreover for any \(\epsilon >0\) the estimate \[ \sup_{x}| F_j(x)-\psi (x)| =O_{\varepsilon}(j^{-1/3+\varepsilon}) \] holds. Next, an explicit expression is given for the convolutions \(\rho_N=Y_1*Y_2*\cdots *Y_N\), \(N=2,3,\ldots\).
Finally, it is shown that both \(1-\psi (x),\) and \(\rho(x)\) , are \(o(\exp (-e^{\{(1/\sigma)-\varepsilon \}\cdot x}))\) for any \(\varepsilon >0\), as \(x\to \infty\).

Reviewer: Wolfgang Schwarz (Frankfurt / Main)

MSC:

11N60	Distribution functions associated with additive and positive multiplicative functions
60F99	Limit theorems in probability theory
11K65	Arithmetic functions in probabilistic number theory

Keywords:

convergence of distribution functions; random variables; infinite convolution