Timeline for Toward a cyclotomic Riemann hypothesis
Current License: CC BY-SA 4.0
19 events
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| Jun 27, 2020 at 13:49 | comment | added | Will Sawin | Yes, I misunderstood what you meant by representative. Your map is well-defined, but I believe the expected asymptotic will be different as soon as the degree is bigger than $1$ (so $n$ not $1,2,3,4,6$.) This is because your quantity is essentially a small power of the norm. You'll get something like $e^{ log n^{ 1-1/d} / \log \log n}$ if I did the math right, where $d$ is the degree. It may not be as sensitive to the Riemann hypothesis in this case. | |
| Jun 27, 2020 at 13:38 | comment | added | Sebastien Palcoux | @WillSawin: I disagree with your sentence stating that $\tau$ (as defined in the post) is not invariant by multiplication by a unit (and then that no invariant representative map exists), because an element of $x \in \mathcal{P}_r$ is precisely an equivalence class of the form $x = a \mathcal{O}$ with $\mathcal{O}$ the group of units, so $x$ itself is already invariant by mutliplication by units: $\forall u \in \mathcal{O}$, $ux = ua\mathcal{O} = au\mathcal{O} = a \mathcal{O} = x$. | |
| Jun 27, 2020 at 13:32 | comment | added | Will Sawin | You would probably count principal divisors using characters of the ideal class group. The contribution of the trivial character would dominate. If not, it's possibly the RH for Hecke L-functions associated to characters of the ideal class group would come into play. | |
| Jun 27, 2020 at 13:14 | comment | added | Sebastien Palcoux | @WillSawin:This post only deals with the principal ideals, whereas the Dedekind zeta function is a sum over all the nonzero (possibly non-principal) ideals. So perhaps it is equivalent to an ERH for the principal part of the Dedekind zeta function. I don't know if such a "principal" ERH is possibly inequivalent to ERH, or if it is even relevant. | |
| Jun 27, 2020 at 13:04 | comment | added | Will Sawin | (1) No map that provides a representative can possibly be invariant under multiplication by a unit (easy to check) so certainly that is not a good idea. (2) Yes, I think it's almost certain that it is equivalent to the RH for the appropriate Dedekind zeta function - just go through the same proof. | |
| Jun 27, 2020 at 13:02 | comment | added | Sebastien Palcoux | @WillSawin: Got it, you certainly mean the absolute norm after identifying $\mathcal{P}_r$ with the set of principal ideals of $\mathbb{Z}[\zeta_r] \cap \mathbb{R}$. This norm maps to $\mathbb{N}$, so does not provide a representative of an element of $\mathcal{P}_r$ (seen as an equivalence class), but seems easier to deal with. Then the mean question is whether the cyclotomic RH defined in this post is equivalent to the extended RH for the corresponding Dedekind zeta function (i.e. does Robin's theorem extended to ERH?). | |
| Jun 27, 2020 at 12:16 | comment | added | Sebastien Palcoux | @WillSawin: No specific reason, this is what came in my mind when I wanted to define a non-negative representative. But I am not number theorist and I am open to improvement. What is the definition of the norm on $\mathcal{P}_r$? | |
| Jun 27, 2020 at 12:02 | comment | added | Will Sawin | Your definition of $\tau$ is extraordinarily complicated. Furthermore, if it is not stable under multiplication by units, then your divisor function sum is not well-defined. Is there a reason to not just use the norm? | |
| Jun 27, 2020 at 11:42 | comment | added | Sebastien Palcoux | @FrançoisBrunault: Thanks! I improved the post according to your comments. I realize that historically the notion of Dedeking ring was specifically created to study the cyclotomic ring $\mathbb{Z}[\zeta_r]$, which is factorial for $1 \le r \le 22$ but not in general (in particular $r=23$). | |
| Jun 27, 2020 at 11:34 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Positive answer of Q1 (after the comment of François Brunault)
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| Jun 27, 2020 at 10:52 | comment | added | François Brunault | @SebastienPalcoux This is true because of uniqueness of factorisation into prime ideals. More generally, this holds in Dedekind rings. | |
| Jun 27, 2020 at 10:44 | comment | added | Sebastien Palcoux | @FrançoisBrunault What (or where) is the proof of that? A chain of dividing ideals (from a nonzero one) is finite because the ring is Noetherian, but how to prove that a nonzero ideal has only finitely many minimal dividing ideals? | |
| Jun 27, 2020 at 9:01 | comment | added | François Brunault | You are (essentially) looking at the ring of integers of the maximal real subfield $\mathbb{Q}(\zeta_r)^+$ of the cyclotomic field. There are only finitely many ideals dividing a given nonzero ideal. | |
| Jun 27, 2020 at 3:17 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
minor edit
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| Jun 27, 2020 at 2:22 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
minor edit
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| Jun 26, 2020 at 13:33 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Modification: as pointed out by Wojowu, the group of unit could be huge, so we quotiented by it and introduce a natural representative of the classes
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| Jun 26, 2020 at 13:33 | comment | added | Sebastien Palcoux | @Wojowu: I just did some modifications according to your comment. Now Q1 asks whether the quotient by the unit group makes the divisor set finite. Next to define the divisor function, and I introduced a natural representative for each quotient class. | |
| Jun 25, 2020 at 15:27 | comment | added | Wojowu | I'm fairly certain the cyclotomic units are dense in $\mathbb R$ for all $r$ a prime larger than $5$, and hence $D(x)$ is going to be very much infinite. This should follow from Dirichlet Unit Theorem and the classical fact that all units are a root of unity times a real number. | |
| Jun 25, 2020 at 15:17 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |