Question

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:

(1) What is the maximum of the set $\{d_i/d_{i-1}: 1\leq i \leq l\}$. Say it $M$.

Assume that you know the prime factorization of $n (= p_1^{\alpha_1}..p_r^{\alpha_r})$. Can I have a formula in terms of $p_i$'s and $\alpha_i$'s. One of the crude upper bound can be $p_r$ but this is really bad if $n$ has many distinct prime factors.

(2) Instead of maximum, the average may be more interesting and useful. So can we estimate the mean and variance,

$\mu = (1/l)\sum_{i=1}^{l}{d_i/d_{i-1}}$ and $\sigma^2 = (1/l)\sum_{i=1}^{l}{(d_i/d_{i-1}-\mu)^2}$

Answer 1

Tenenbaum [Sur un probleme de crible et ses applications (1986)] showed that $$ F(n)/n=\max_{1\le i < \tau(n)} \frac{d_{i+1}(n)}{d_i(n)},$$ where $1=d_1(n)< \ldots < d_{\tau(n)}(n)=n$ is the increasing sequence of divisors, and $F$ is given by $F(1)=1$ and $F(n)=\max \{ d P^{-}(d) : d|n,\, d>1 \}$ for $n\ge 2$, where $P^{-}(n)$ is the smallest prime divisor of $n$. For $n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, with $p_1<\ldots < p_k$, this leads to the formula $$ \max_{1\le i < \tau(n)} \frac{d_{i+1}(n)}{d_i(n)} = \max_{1 \le j \le k} \{ p_j p_j^{\alpha_j}\cdots p_k^{\alpha_k} \}/n$$

Let $D(x,t)=|\{n\le x: F(n)/n \le t\}|$. I showed [Integers with dense divisors, 2 (2004)] that $D(x,t)=x\, d(w) \{1+O(1/\log t) \}$, for $x\ge 3$, $x\ge t \ge \exp\{(\log(\log(x)))^{5/3+\varepsilon}\}$, where $w=\log x / \log t$ and $d(w)$ is a continuous function that can be expressed in terms of Dickman's function and which satisfies $d(w) \asymp 1/w$.