Timeline for Update for 2015: least prime of form nq+1, with q prime?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2015 at 4:17 | comment | added | Vesselin Dimitrov | I see - as in Tsfasman and Vladut's paper Infinite global fields and generalized Brauer-Siegel theorem, if I understand this correctly, which improves the Odlyzko bounds under additional splitting conditions at finite primes. Thank you! (By the way, above I made a pair of cancelling sign errors. The Kronecker invariant should be the constant term of $\zeta_K'(s)/\zeta_K(s)$ at $s = 1$, and its positivity would be one way to quantify the deficiency of small split primes.) | |
| Sep 11, 2015 at 3:17 | comment | added | Lucia | It wasn't so precise a remark, but if you look at the Odlyzko bounds for discriminants, then if there is a small prime that splits completely then one gets an improvement of those bounds. Thus for minimal discriminants not too far from the Odlyzko bounds, one usually has that most of the small primes are not split. | |
| Sep 11, 2015 at 3:14 | comment | added | Vesselin Dimitrov | Is this something easy to explain in a few lines, or is there a handy reference? For instance, will a sequence of fields with bounded root discriminant and degrees going to infinity (such as an infinite unramified tower), never contain primes of small norm? Will the Euler-Kronecker constants (the constant terms of $-\zeta_K'(s)/\zeta_K(s)$ at $s = 1$) eventually all be negative? (Or perhaps I should make this into a different question?) The negativity of these quantities is related to the lack of primes of small norm; would they nearly achieve $-\log{\sqrt{D}}$ when $D$ is unusually small? | |
| Sep 11, 2015 at 3:04 | comment | added | Vesselin Dimitrov | "Discriminants of number fields are small precisely when there are no primes of small norm" - I am not sure I see this outside of the setting of Lehmer's problem (which, indeed, concerns WLOG only/precisely the number fields of log discriminant $\ll d\log{d}$ and no unramified primes of norm $< d\log{d}$. Perhaps the qualifier "unramified" here could be also dispensed with.). For polynomials this is clear, just because a prime of small norm would have to divide the discriminant highly, but the fields say of bounded root discriminant will most likely never be monogenic. | |
| Sep 10, 2015 at 22:29 | comment | added | Lucia | @VesselinDimitrov: Thanks for those remarks. It seems to me that the lower bounds for Mahler measures precisely corresponds to the case of large degree and small discriminant. I don't really have a good intuition for that case -- and one also knows that discriminants of number fields are small precisely when there are no primes of small norm. It would be interesting to figure out a uniform such conjecture, but I don't know offhand what to expect. | |
| Sep 10, 2015 at 21:51 | comment | added | Vesselin Dimitrov | And second, Mahler's inequality (interpreting a discriminant as the square of a Vandermonde determinant) bounds the absolute logarithmic discriminant of that field (indeed the discriminant of the minimum polynomial of $\alpha$) by $d \log{d} + 2d(d-1)h(\alpha)$. But we can assume $h(\alpha) < 1/d$ in this problem, and the optimistic hypothesis would give us a small prime $p$ to use in the first observation. | |
| Sep 10, 2015 at 21:49 | comment | added | Vesselin Dimitrov | We would need the degree to go to infinity in this kind of application. Actually what I had in mind is simpler than Dobrowolski's argument, and consists of combining two observations. First, if $\alpha$ is an algebraic unit not a root of unity and $p$ a split prime in $\mathbb{Q}(\alpha)$, then $\xi := \alpha^{p-1} - 1$ is a non-zero algebraic integer having positive valuation at all places dividing $p$; applying the product formula to $\xi$ gives $h(\alpha) \geq (\log{(p/2)})/(p-1)$. (If $p = 2$, do the same with $\alpha^3-1$ instead.) | |
| Sep 10, 2015 at 21:04 | comment | added | Lucia | @VesselinDimitrov: That's an interesting observation, also consistent for example with what we believe about the least quadratic non-residue. Do you have in mind that the degree of the number field is fixed and discriminant goes to infinity, where I would believe a heuristic of this type; or do you have in mind that the degree can go to infinity as well (which is the cyclotomic case, but at least the discriminant is pretty big), where I would be more cautious (e.g. with minimal discriminants)? (Dobrowolski's argument is not so fresh in my mind that I can see what you're thinking.) | |
| Sep 10, 2015 at 20:48 | comment | added | Vesselin Dimitrov | In terms of the discriminant $\Delta$ of the cyclotomic field $C_q$ the optimistic conjecture is expressed (in slightly less precise form to fix ideas) as a bound $\ll \log{\Delta} \cdot (\log{\log{\Delta}})^{1+o(1)}$. I wonder if anything has been said about the literal extension of this to a hypothetical bound on the first split prime for a general number field $K$, where now $\Delta = |D_{K/\mathbb{Q}}|$? If true, this would break Dobrowolski's bound on the Lehmer problem and yield $h(\alpha) \gg d^{-1} \cdot (\log{d})^{-1-o(1)}$. Is there a known obstruction in this generality? | |
| Sep 10, 2015 at 19:19 | vote | accept | Will Jagy | ||
| Sep 10, 2015 at 18:15 | comment | added | Lucia | It's just the usual reasoning: the "probability" of $kq+1$ being prime is about $q/(\phi(q)\log q)$ (let's just say about $1/\log q$ for simplicity), and then you use a "Borel-Cantelli" heuristic (and arrive at the conjecture of $\ll \phi(q) (\log q)^2$). I don't know if anyone stated exactly this -- Granville and Pomerance did conjecture in a paper that sometimes the least prime should be $\gg \phi(q) (\log q)^2$ (more precisely, for each large $q$ they conjectured that there is some progression for which the least prime is that large). | |
| Sep 10, 2015 at 18:03 | comment | added | Will Jagy | Lucia, do you think there is anything in print that, say, sketches how Cramer's general viewpoint suggests $q \log^2 q?$ | |
| Sep 10, 2015 at 17:33 | comment | added | Will Jagy | That's a good point, and not something I would have figured out, that the $q \log^2 q$ I found numerically is related to Cramer's rather than GRH. | |
| Sep 10, 2015 at 17:26 | history | answered | Lucia | CC BY-SA 3.0 |