Timeline for Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
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7 events
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| Mar 18, 2015 at 17:00 | comment | added | GH from MO | @Slamice: I don't think you can prove along these lines that the sum of reciprocals of twin primes is finite. That is, I don't think there is a direct proof that the span of twin primes has zero density. Brun's proof estimates from above the number of twin primes up to $x$ (which is a finer information than the convergence of the sum of reciprocals), and I don't think it gets simpler than that. In fact it is not too natural to look at the reciprocal sum of twin primes or their span. What is important is the density of twin primes among all primes, and this is what Brun addressed. | |
| Mar 18, 2015 at 8:44 | comment | added | Slamice | Ok, not a really fine information, but it's much simpler than most of the proofs of the (of course already simple) classical statement $\sum_{p\in \mathbb{P}}1/p=\infty$, isnt it? And: It reveals the "real reason" for this statement, whre the pure manipluation analytics arguments with logarithms does not. Also it should be not too hard to understand, that the span of the prime twins set has zero densitiy ... I expect that this would be an easier approach too bruns theorem (without the estimating of course, but only the convergence). | |
| Mar 17, 2015 at 21:51 | comment | added | Anthony Quas | My guess is that so crude that it's unlikely there is really fine information hidden there. I'm not at all an analytic number theorist so can't really comment further. | |
| Mar 17, 2015 at 20:07 | vote | accept | Slamice | ||
| Mar 17, 2015 at 20:07 | comment | added | Slamice | To add more information how this question came up: I noticed that Erdös used only property of $\mathbb{P}$ in his proof of the divergence of the sum of the reciprocals of $\mathbb{P}$, so the generalization was quite near. Is this criterion ever used in any context, or is it just to trivial to be useful? | |
| Mar 17, 2015 at 20:07 | comment | added | Slamice | Thanks, two questions to that: So now we have a simple characterization criterion for a subseries of the harmonic sum to converge/diverge, especially a very easy way to see the divergence of the sum of reciprocels of $\mathbb{P}$. Why this criterion is not the usual one to teach and learn, has it a name? And does it give some possibility to reformulate or simplify the proof of the existence of brun's constant? | |
| Mar 17, 2015 at 17:18 | history | answered | Anthony Quas | CC BY-SA 3.0 |