Timeline for Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]
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12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2015 at 19:38 | vote | accept | Mostafa - Free Palestine | ||
| Mar 8, 2015 at 18:09 | history | closed |
user9072 Lucia Boris Bukh Stefan Kohl♦ Alex Degtyarev |
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| Mar 8, 2015 at 14:48 | comment | added | GH from MO | I don't know what you mean by "directly". Mertens' theorem readily gives the sum of the error terms. The idea of sieve theory is to introduce weights that cut down the number of terms in the sieving process. This is also how Zhang proved his spectacular theorem that there are infinitely many prime pairs with distance at most 70 million apart. Since then, 70 million was lowered to 246. This is a pretty good approximation to the twin prime conjecture, although still very far away from it. Number theory is hard. | |
| Mar 8, 2015 at 14:44 | comment | added | Mostafa - Free Palestine | @GHfromMO Does there exist a way to work directly with these error terms? | |
| Mar 8, 2015 at 14:39 | comment | added | GH from MO | As MrSelberg explained below, your conjectured heuristics is false. The reason is, very briefly, that there are two many terms in your sums and error terms pile up. This is the main difficulty to deal with in sieve theory in general! | |
| Mar 8, 2015 at 13:04 | review | Close votes | |||
| Mar 8, 2015 at 18:09 | |||||
| Mar 8, 2015 at 8:36 | comment | added | Mostafa - Free Palestine | @EmanueleTron The main reason for the question is that one can easily relate the number of twin primes less than n to a similar sum and if we have good estimates on the fractional parts, their density can be related to $\frac 12 \prod_{2< p\leq \sqrt{x}} (1-2p^{-1})$ . | |
| Mar 8, 2015 at 8:30 | comment | added | user41593 | You asked "Is this formula true? If so, can this be completed to an exact proof?": the premise about the formula is false, so no point worrying of turning a heuristic for a wrong result into a rigorous proof! Anyway, what bothers me the most is that you use the Prime Number Theorem in the very first asymptotic equivalence and still you worry about not using it in the follow-up. | |
| Mar 8, 2015 at 8:24 | comment | added | Mostafa - Free Palestine | Nice! But I need to know how to handle the fractional parts in the first sum. | |
| Mar 8, 2015 at 8:20 | answer | added | MrSelberg | timeline score: 7 | |
| Mar 8, 2015 at 8:14 | comment | added | user41593 | The constant is wrong. See en.wikipedia.org/wiki/Mertens%27_theorems | |
| Mar 8, 2015 at 8:05 | history | asked | Mostafa - Free Palestine | CC BY-SA 3.0 |