Timeline for What are the limits of the Erdős-Rankin method for covering intervals by arithmetic progressions?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2012 at 2:37 | comment | added | Junkie | Maier-Pomerance indicate that they expect $z(\log z)^2$ as the limit (see 1.5), if one knew prime $k$-tuples. They basically use an on-average version of that (in AP), in the paper improving the constant. Thus for large primes, they can't show that any of them individually sieves out more than 1 number, but on average they can show at least 1.31, and Pintz 2. When knowing prime $k$-tuples, at least with uniformity enough, the large primes would then be shown to be more optimal, in sieving out. | |
| Aug 26, 2012 at 3:15 | comment | added | Douglas Zare | It looks like Iwaniec proved $O(z^2)$ in Iwaniec "On the Problem of Jacobsthal" Demonstratio Math. 11 (1978) 225-231. | |
| Aug 26, 2012 at 0:22 | comment | added | Douglas Zare | I wouldn't be shocked if it could be improved by a really clever trick, but I'd still like to know if there is some clear obstruction to improving it all of the way to $z^2$, say. If so, then to prove there are large prime gaps one has to use other techniques than covering intervals by arithmetic progressions, although that still looks like a natural problem on its own. @Gerhard, I'm not very familiar with the recent progress on this problem, and I'll look into the work you mention. Thanks. | |
| Aug 25, 2012 at 23:58 | comment | added | Woett | If I remember correctly, Erdös himself was positive that this construction can't be improved easily (he called it hopeless even), which is the reason he offered a large prize for it. | |
| Aug 25, 2012 at 22:18 | comment | added | Gerhard Paseman | Also, Westzynthius uses a similar argument to get bounds close to what Rankin and Erdos have. I will review the paper and post something summarizing the differences between W's method and the one you outline above (which may very well be no difference). Gerhard "Ask Me About System Design" Paseman, 2012.08.25 | |
| Aug 25, 2012 at 21:46 | comment | added | Gerhard Paseman | I am still working through the literature myself, so I don't know the answer. I take it you know of the further advances on prime gap lower bounds (Pomerance, Maier, Pintz, I think?), and that they bear no resemblance to Rankin's method? Also, have you checked Hagedorn's 2009 paper on computing Jacobsthal's function to make sure there is nothing you want there? Gerhard "Just Checking On The Obvious" Paseman, 2012.08.25 | |
| Aug 25, 2012 at 19:51 | history | asked | Douglas Zare | CC BY-SA 3.0 |