Distribution function for divisors of an Integer - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:28:26Z http://mathoverflow.net/feeds/question/115496 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115496/distribution-function-for-divisors-of-an-integer Distribution function for divisors of an Integer Kamalakshya 2012-12-05T12:50:08Z 2012-12-05T13:58:11Z <p>For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$, where $p$ is the smallest prime factor of $n$.</p> <p>For example if $n = p^i$ for some prime $p$ then $D_n(x) \sim \log x/ \log p$. </p> <p>What are the other 'nice' distribution function of divisors that are satisfied by an infinite family of integers ( like $\log x/ \log p$ for $\{p^i, i \in \mathbb{N}\}$ ).</p> <p>Here 'nice', means it is easy to do calculus (integrate or differentiate) with such functions. </p> http://mathoverflow.net/questions/115496/distribution-function-for-divisors-of-an-integer/115500#115500 Answer by quid for Distribution function for divisors of an Integer quid 2012-12-05T13:58:11Z 2012-12-05T13:58:11Z <p>The distribution of divisors on 'average' is known (so to say giving the behavior of a 'typical'integer).</p> <p>Set <code>$F_n(u) = d(n)^{-1} D_n(n^u)$</code> where $d(n)$ is the total number of divisors of $n$, then </p> <p>$$x^{-1} \sum_{n\le x} F_n(u) = \frac{2}{\pi} \arcsin \sqrt{u} + O((\log x)^{-1/2})$$ uniformly for $x\ge 2$ and $u \in [0,1]$. This is due to Deshouillers, Dress, Tennenbaum (1979) and can be found for example in Tennenbaum's Introduction to Analytic and Probabilistic Number Theory (see Thm 7 in II.6). </p> <p>Roughly this means, there are many small divisors. </p> <p>A lot of information on this type of question, and I assume also this result but could not check, is in Hall and Tennenbaum's book 'Divisors'. </p>