Timeline for Bateman-Horn conjecture, continued
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 27, 2015 at 14:35 | comment | added | Vesselin Dimitrov | @BorisBukh: That $n_p = 1$ on average is nothing but Landau's prime ideal theorem for non-normal fields, which is very much simpler than Chebotarev. And here, for the convergence at hand, this is only needed in logarithmic form (Dirichlet rather than natural density), which by a straightforward arrangement amounts to establishing that the Dedekind zeta function has a simple pole at $s = 1$ (and can even be arranged in simple terms not mentioning the zeta function). | |
| Aug 16, 2015 at 14:46 | comment | added | joro | @IgorRivin Why care about sign change? From wikipedia: "They, however, do have a logarithmic density, which is approximately 0.9959". This might not be enough for answer of course. | |
| Aug 16, 2015 at 14:44 | comment | added | Igor Rivin | @joro Chebyshev bias is not strictly speaking true (the sign of the difference does change). | |
| Aug 16, 2015 at 14:42 | comment | added | joro | Two things bug me. First, one can make the constant arbitrary large rational: take $f(x)=primorial(k)x+1$ (infinitely many terms vanish). Second, there is Chebyshev bias which states "there are more primes of the form $4k + 3$ than of the form $4k + 1$, up to the same limit". So large enough Chebyshev bias might be a hint... | |
| Aug 16, 2015 at 14:12 | comment | added | Igor Rivin | @FedorPetrov See my answer, though I don't claim to understand what is going on. | |
| Aug 16, 2015 at 14:12 | answer | added | Igor Rivin | timeline score: 2 | |
| Aug 16, 2015 at 13:50 | comment | added | Igor Rivin | @BorisBukh The average of $n_p$ is $1$ when the Galois group is the whole $S_d,$ but it's far from clear for other Galois groups (at least to me). That's part of the reason why I asked the question: from the Chebotarev argument, it is far from clear that the product is always finite. | |
| Aug 16, 2015 at 12:28 | comment | added | Fedor Petrov | Wiki articles claims that this product is always positive (without reference), I guess it is also always finite. This is a statement of Chebotarev Density Theorem type, though quite strong. | |
| Aug 16, 2015 at 12:26 | comment | added | Boris Bukh | I really wish that the answer to the first question turns out to be a "no". I would love to see a proof! (Seriously, one probably cannot avoid Chebotarev's density theorem, for if the average of $n_p$ is not $1$, the product would diverge.) | |
| Aug 16, 2015 at 12:12 | history | asked | Igor Rivin | CC BY-SA 3.0 |