Timeline for What is the growth rate for divisibility of integers
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2012 at 20:17 | history | edited | Charles |
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| Jan 1, 2012 at 18:49 | vote | accept | David Spivak | ||
| Jan 1, 2012 at 0:49 | answer | added | Will Jagy | timeline score: 7 | |
| Jan 1, 2012 at 0:17 | comment | added | Frank Thorne | If you don't count with multiplicity then this is a standard problem and $E(N)$ is asymptotic to $\log \log N$. This is proved in the beginning of Tao and Vu's book, among (I presume) many other places. If you count with multiplicity, I anticipate that you will get at least $O(\log \log N)$ and probably an asymptotic, although I didn't work out the details. | |
| Jan 1, 2012 at 0:15 | comment | added | Timothy Foo | Hope you don't mind me switching notation from $PF(n)$ to $\Omega(n)$. :) $\Omega(n) \leq \tau(n)$, the number of divisors of $n$, and $\sum_{n \leq x}\tau(n) \sim x \log x$ by a result of Dirichlet, so this gives what you want. | |
| Jan 1, 2012 at 0:14 | comment | added | Gerhard Paseman | Most integers n have about loglogn factors, which when averaged will make little difference whether multiplicity is considered. I suspect a web search on the phrase "number of factors" may be enlightening. Also, this subject should be covered in a handbook on number theory as well as many analytic number theory texts. Not being a number theorist, I will leave naming the texts to others. Gerhard "But I Do It Anyway" Paseman, 2011.12.31 | |
| Jan 1, 2012 at 0:12 | answer | added | David Spivak | timeline score: 0 | |
| Jan 1, 2012 at 0:00 | history | asked | David Spivak | CC BY-SA 3.0 |