Timeline for Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30 at 9:17 | vote | accept | Alexandre | ||
| Aug 28 at 3:32 | history | edited | Michael Hardy | CC BY-SA 4.0 |
deleted 11 characters in body
|
| Aug 27 at 15:51 | answer | added | Fedor Petrov | timeline score: 23 | |
| Aug 27 at 13:45 | comment | added | Alexandre | This is precisely my interrogation. The Farey enumeration (with inverses) of order $n^2$ with weights $\frac{1}{n\sqrt{ab}}$ converges to another measure $e^{-\frac{1}{2}|x|}\textrm{d}x$. | |
| Aug 27 at 13:40 | comment | added | Iosif Pinelis | @Alexandre : Thank you for the reference. However, there the condition on $a,b$ seems to be $\max(a,b)\le n$, whereas your condition is $ab\le n$. Not sure if this difference is significant. | |
| Aug 27 at 13:29 | comment | added | Alexandre | "An Introduction to the Theory of Numbers" from Hardy & Wright (section 18.5) speaks about the density $1/\zeta(2)$. | |
| Aug 27 at 13:27 | comment | added | Iosif Pinelis | @FedorPetrov : "points with coprime coordinates being uniformly distributed" Can you give a reference to this fact? | |
| Aug 27 at 13:24 | comment | added | Alexandre | It is probably related to this uniformity in a way or another, but the detail in not clear to me. $\zeta(2)$ can be factored out of a more brute expression $\sum_{k,l=0}^\infty \frac{F(\log \frac{k}{l})}{kl}$ (which does not converge). The fact that we have to trim the rationals with $ab>n$ make the enumeration of rationals different from (but included in) the union of the Farey sequence and the set of inverses, up to order $n^2$. | |
| Aug 27 at 10:09 | comment | added | Fedor Petrov | It is less or more equivalent to points with coprime coordinates being uniformly distributed on the plane with density $1/\zeta(2)$ (that is well known), thus, for smooth enough functions the answer is positive. For example, this holds for the class suggested by fedja (which looks reasonable for me and usual in such results). | |
| Aug 27 at 8:29 | history | edited | Alexandre |
edited tags
|
|
| Aug 27 at 8:19 | comment | added | Alexandre | Yes, I have introduced $F$ as a test function here, to formulate the question about the underlying empirical distribution, which is seemingly asymptotically uniform and equal to 1. I wonder how close this measure is to the Lebesgue measure. | |
| Aug 27 at 2:48 | comment | added | fedja | Is Riemann integrable with compact support sufficient for your purposes? | |
| Aug 26 at 15:45 | history | asked | Alexandre | CC BY-SA 4.0 |