Timeline for Chebotarev density theorem for $k$-almost primes
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2014 at 18:21 | vote | accept | Joël | ||
| Feb 8, 2014 at 1:28 | comment | added | Lucia | @alpoge: Thanks; I deleted my answer as I realized that Joel's question had an extra subtlety. This is now addressed in the answer below. For the Selberg-Delange method see Tenenbaum's book on analytic and probabilistic number theory; or Selberg's original paper (Note on a result of L.G. Sathe). | |
| Feb 8, 2014 at 0:49 | comment | added | alpoge | @Lucia: great answer! (--- though it seems to have been deleted now.) But what is the Selberg-Delange method? Pardon my ignorance! I just couldn't find a decent reference when Googling. | |
| Feb 8, 2014 at 0:30 | answer | added | Lucia | timeline score: 11 | |
| Feb 7, 2014 at 20:43 | history | edited | Joël | CC BY-SA 3.0 |
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| Feb 7, 2014 at 20:15 | comment | added | Joël | Also, one reason to ask the question for a fixed $k$ is that it is possible that it has a simple answer, like: "yes, your expected equivalent for $A(x)$ is true with this value for the constant $c$", perhaps even the naive $c=|D|/|G|^k$; or "no; your formula is not true for any value of $c$". | |
| Feb 7, 2014 at 20:13 | comment | added | Joël | @Lucia. I am interested in $k$ large. Actually the real thing I need is somewhat different: I have a fixed $G$, and a family of sets $D_k \subset G^k$ for all $k$, hence a set $A_k$ of integers for all $k$ as in the question, and I am interested in the density of the set $A=\cup_k A_k$. I shied away of asking this general question, restraining myself to the individual $A_k$, even though I know that it will be far from easy to deduce, if at all possible, to deduce a result for $A$ from results from the $A_k$. If you know of a more direct route to $A$, I'm very interested. | |
| Feb 7, 2014 at 18:26 | comment | added | Lucia | If $k$ is fixed and small, you can obtain such a result using Chebotarev and partial summation (painful but standard). If you are interested in $k$ maybe being large, one would have to work out the analog of Selberg's work. Also one can adapt the answer I gave to the problem of counting $k$-almost primes up to $x$ in order to get precise asymptotic formulae (e.g. on GRH). | |
| Feb 7, 2014 at 18:24 | history | edited | Joël | CC BY-SA 3.0 |
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| Feb 7, 2014 at 18:19 | history | asked | Joël | CC BY-SA 3.0 |