Timeline for Which kind of subsets of primes one needs to generate a positive ratio of the natural numbers?
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15 events
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| Mar 17, 2015 at 21:49 | comment | added | Anthony Quas | I'd say this is probably reasonably well known. Ideas very much like this show up in Mertens' Theorem. Also some morally similar ideas occur in my paper math.uvic.ca/faculty/aquas/papers/paper38.pdf | |
| Mar 17, 2015 at 21:13 | comment | added | Slamice | Thanks for your proof @AnthonyQuas: Is this criterion known, what is its name? | |
| Mar 17, 2015 at 20:07 | vote | accept | Slamice | ||
| Mar 17, 2015 at 18:20 | comment | added | Richard Stanley | @GeoffRobinson: yow! I meant congruent to 2 mod 4, not 0 mod 4. | |
| Mar 17, 2015 at 17:18 | answer | added | Anthony Quas | timeline score: 12 | |
| Mar 17, 2015 at 17:07 | comment | added | Anthony Quas | I think you're right. The condition should be the sum of the reciprocals of $\mathbb P\setminus S$ should be finite. I strongly suspect this is necessary and sufficient. | |
| Mar 17, 2015 at 16:29 | comment | added | Geoff Robinson | @RichardStanley : I think there may be typo in your comment?? | |
| Mar 17, 2015 at 16:28 | comment | added | Gerhard Paseman | Erik Westzynthius was a Finnish actuary whose major contribution to mathematics was a 1931 paper on a sieving process. This process showed that for any constant C, there were infinitely many primes p_n such that p_{n+1} > p_n + Clog p_n. Before then, no one knew prime gaps could get much larger than average. Gerhard "Terry Tao's Blog Has More" Paseman, 2015.03.17 | |
| Mar 17, 2015 at 16:07 | comment | added | Richard Stanley | If $S$ consists of primes congruent to 0 or 1 mod 4, then the density is 0. This is because every element of span$(S)$ is a sum of two squares, and the set of positive integers that are a sum of two squares has density 0. In fact, we can throw into $S$ the squares of all primes congruent to 3 mod 4, and the density is still 0. | |
| Mar 17, 2015 at 15:45 | comment | added | Slamice | What about some density-criterions like the serial of the reciprokes, which may converge oder diverge ... is there a known connection? | |
| Mar 17, 2015 at 15:43 | comment | added | Vincent | So... Who is Erik Westzynthius? | |
| Mar 17, 2015 at 15:42 | comment | added | Gerhard Paseman | It may develop that you are after zero-density subsets of the primes, where there is some already developed notion of density. You might search this forum for the word density with some number theory tag attached. Gerhard "The Answer May Be Close" Paseman, 2015.03.17 | |
| Mar 17, 2015 at 15:38 | comment | added | Gerhard Paseman | Note that the set is related to the totatives of finite products of members outside of S. In particular, if the complement of S is a finite set T, the product of the members of T is then called n, then S is related to the totatives of n, and your epsilon can be chosen not much smaller than a constant times 1/log(log(n)). The next thing to look at would be S having only those primes congruent to b mod d for some integers b coprime to d. Gerhard "Ask Me About Erik Westzynthius" Paseman, 2015.03.17 | |
| Mar 17, 2015 at 15:30 | review | First posts | |||
| Mar 17, 2015 at 15:41 | |||||
| Mar 17, 2015 at 15:28 | history | asked | Slamice | CC BY-SA 3.0 |