Timeline for Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 30 at 12:54 | comment | added | Alexandre | The convergence worsens a lot for any $\gamma$ if summing over all integer points below the hyperbola (without the factor $\zeta(2)$). It looks surprisingly good only when summing over coprimes for $\gamma=\frac{1}{2}$. It may have something to do with the conjecture I have posted there (which may be more suitable for MO...)? | |
| Aug 30 at 12:05 | comment | added | Fedor Petrov | Is it faster also for the summation over all integer points, not necessarily coprime? | |
| Aug 30 at 11:58 | comment | added | Alexandre | As a consequence of your argument, we can also notice that any weight of the form $2(1-\gamma)\zeta(2)(ab)^{-\gamma}n^{\gamma-1}$ should work just as fine. However, I observe that the convergence is faster for $\gamma=\frac{1}{2}$. Any heuristic for that? | |
| Aug 30 at 10:52 | comment | added | Fedor Petrov | Oops, it does not converge of course, I missed several restrictions, now added | |
| Aug 30 at 10:51 | history | edited | Fedor Petrov | CC BY-SA 4.0 |
added 19 characters in body
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| Aug 30 at 9:17 | vote | accept | Alexandre | ||
| Aug 30 at 9:17 | comment | added | Alexandre | Thank you for the additional hints. I still find the details of the last paragraph slightly confusing, like the relation between $\Omega_{s,r,c}$ or the integral $\int_{x,y>0}\frac{dxdy}{\sqrt{xy}}$ which in fact does not converge. But the argument about the uniform distribution of coprime and the last integral teached me a lot! | |
| Aug 28 at 11:27 | comment | added | Emil Jeřábek | I see, thank you. | |
| Aug 28 at 9:16 | comment | added | Fedor Petrov | @EmilJeřábek for fixed $b\leqslant \kappa n$ the number of $a\leqslant n$ coprime to $b$ is $n\cdot \frac{\varphi(b)}b+o(n)$, then sum up over $b$. See, for example, the reference here mathoverflow.net/a/22959/4312 | |
| Aug 28 at 9:12 | comment | added | Emil Jeřábek | I guess I am being dense, but can you elaborate on your last comment? The proof I am familiar with does not seem to readily generalize to $a\le N$, $b\le\kappa N$. It starts with $\sum_{\substack{a,b\le N\\(a,b)=1}}1=2\sum_{\substack{b\le a\le n\\(a,b)=1}}1-1=2\sum_{a\le N}\varphi(a)-1$; how do you modify these steps to accommodate $\kappa\ne1$? | |
| Aug 28 at 4:39 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting
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| S Aug 27 at 20:07 | history | suggested | Alexandre | CC BY-SA 4.0 |
Obvious typo, also highlighting a crucial point.
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| Aug 27 at 20:06 | comment | added | Fedor Petrov | You simply approximate (from below and ftom above) the function $1 /\sqrt{xy} $ on the set $\Omega_c$ by a finite sum of characteristic functions or rectangles, reducing the uniform distribution claim to the rectangles $a\leqslant N$, $b\leqslant \kappa N$ with fixed $\kappa$ and large $N$. For such rectangles the proof is the same as for the squares (Moebius inversion in disguise). | |
| Aug 27 at 19:46 | review | Suggested edits | |||
| S Aug 27 at 20:07 | |||||
| Aug 27 at 18:38 | comment | added | Alexandre | Nice explanation! I am trying to understand to point of splitting the sum as you propose. Can't we simply skip this step? The final integral explains well the role of the condition $ab<n$. Also, this holds assuming that the coprime points are indeed uniformly distributed in the plane. However, the reference I have found (Hardy & Wright, section 18.5) only states the average density in a square $\max(a,b)\leq n$. A more precise reference would complete nicely your argument. | |
| Aug 27 at 16:12 | comment | added | Fedor Petrov | @DavidESpeyer change of variables $x=\tau^2$, $y=\xi^2$ also works smoothly | |
| Aug 27 at 16:09 | comment | added | David E Speyer | For those who are wondering how to do the last integral, substitute $x = e^t$, $y = e^u$, then the integrand is $e^{(t+u)/2}$ and the bounds are $p < t-u < q$, $t+u < 0$. Then rotate coordinates: $v = (t+u)/2$, $w= (t-u)/2$. | |
| Aug 27 at 15:51 | history | answered | Fedor Petrov | CC BY-SA 4.0 |