Timeline for Comparing sizes of sets of natural numbers
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2016 at 16:51 | vote | accept | James Propp | ||
| Sep 5, 2016 at 18:14 | comment | added | James Propp | Lucia is correct; I mis-characterized her reply. In any case, my search for information is fairly broad, so other regularization schemes would interest me as well (especially theorems that assert equality between the answers obtained from different regularization procedures). | |
| Sep 5, 2016 at 18:07 | comment | added | Lucia | I don't think I'm pointing out the zeta function regularization (although of course one could study that too) -- in my answer I was describing just the Abel regularization of the original question. | |
| Sep 5, 2016 at 17:59 | comment | added | James Propp | Put $|S|_s = \sum_{n \in S} n^{-s}$ (where $S$ is a set of positive integers). Then Lucia is pointing out that the behavior of $|S|_s - |T|_s$ as $s \rightarrow 0$ is also a natural way to measure the difference in size between $S$ and $T$. When $S=\{1,3,5,\dots\}$ and $T=\{2,4,6,\dots\}$, one has $|S|_s - |T|_s \rightarrow 1/2$ as $s \rightarrow 0$ (see en.wikipedia.org/wiki/Dirichlet_eta_function); I would be interested in knowing what zeta-regularization yields for the other pairs of sets described in the original post. | |
| Sep 4, 2016 at 4:31 | history | answered | Lucia | CC BY-SA 3.0 |