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Jan
30 |
awarded | Good Answer |
Apr
9 |
awarded | Yearling |
Jun
24 |
comment |
On Deligne's determinant of motives
The main paper here (section 3.5) is on genus 2, see eprints.maths.ox.ac.uk/259/1/art15.pdf $$ $$ Another paper, Stoll and Yang go thru a computation for $y^2=x^5+A$, which is special as it has a Hecke character. They never give the period integrals specifically, saying only that it is a "real period" in BSD analogue. With Hecke, might be the only cases that are known (following Blasius). mathe2.uni-bayreuth.de/stoll/papers/StollYang-2002-09-18.pdf |
Jun
19 |
comment |
How is “large” defined in an equality for the modulus of Riemann zeta?
"Large" $t$ in practice, for this example probably means large enough for $\log\log t$ to be non-negligible. |
Jun
12 |
comment |
Does every equivalence class of Hecke characters contain a distinguished element?
Doesn't Weil have some type "A0" of Hecke characters, which are the "algebraic" ones? They include other infinite ones, the Grossencharacters, in CM extensions (see 2nd link below). They do not include those used in XV of Lang (involving a log of a fundamental unit in a real quadratic field, to trivialize units) as in this question mathoverflow.net/questions/65059/… See also mathoverflow.net/questions/111851/… and |
Jun
10 |
awarded | Nice Answer |
Jun
9 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I guess what I meant by avoiding Kloostermania would mean an easier read for a wider audience, was that the manipulations needed in BFI in preparation to applying Deshouillers/Iwaniec are themselves a bit ponderous. Though upon further review, perhaps Zhang has to do just as much gyrations to be able to apply Deligne (or actually Weil in all but the last Section) to the Kloosterman sum. Maybe it is just my taste speaking. |
Jun
9 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Prior to last week, I only looked at BFI quite briefly (and superficially) when I was a grad student, and then in the wake of PRIMES in P, when asked whether the initial AKS result was effective (relying (weakly?) on BFI somehow). The main BFI thrust is their 3 different bounds on $S_1$ sums in Sections 8-10, involving assorted Cauchy reorganizations depending on interval sizes. In all cases (including special cases in 11-13), they appeal to a variant of Lemma 1, which is the Deshouillers/Iwaniec hammer following Kuznetsov. So I guess you can say that, the black box is either Deligne or this. |
Jun
9 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
To be complete, the other "new" elements by Zhang were reducing to smoothed moduli (already in Motohashi/Pintz, and certainly standard in the field), and the handling (Section 10) of more than one residue class per modulus, via a multiplicative system. Again as Kowalski spoke, it was not immediately clear to experts how Zhang would resolve the second issue, as higher Kloostermania is almost forced to consider but a solitary fixed residue. But that is unneeded here (and indeed, is one aspect that keeps the paper readable to a wider audience -- assuming you import Deligne's bounds, of course). |
Jun
9 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
I might say, Zhang could call his paper "A new bound for an incomplete Kloosterman sum to composite moduli, and bounded prime gaps". Kowalski noted (and was kinda surprised) the "higher" Kloostermania such as Kuznetsov trace formula and sums of Kloosterman sums did not appear (as in BFI). To state the "philosophy" of Zhang's new element in one (long) sentence: When you Weyl shift a suitable incomplete Kloosterman sum to a composite modulus by multiples of a (small) factor of said modulus, you beat Deligne since the shifts modulo this factor induce an essentially bounded Ramanujan piece. |
Jun
8 |
comment |
Primes in short intervals with a preassigned frobenius
I think the first result in AP was: K. Prachar, Über den Primzahlsatz von A. Selberg, Acta. Arith., 28 (1975), pp. 277–297. eudml.org/doc/205389 |
Jun
6 |
awarded | Commentator |
Jun
6 |
comment |
Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?
The work of Watkins (or Arno et.al.) can be minutely simplified for $(Z/3)^3$. This is by the "structure of minima" they use. You cannot have a form of order $>3$, and forms of odd order pair conjugately, so every nonprincipal form $(a,b,c)$ must have $a^{(3+1)/2}\ge \sqrt{d/4}$ rather than $a^{(27+1)/2}\ge\sqrt{d/4}$ as with general order 27. I don't know how much this eases the situation, but the papers use similar facts to reduce the sieving problems in their cases. I would not be surprised if handling $(Z/3)^k$ for $k=5$ or even more is feasible. Even order case is much more difficult. |
Jun
6 |
comment |
Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?
This is wrong, the hard part of Watkins's thesis wasnot the range near $e^{5077h}$, but in the reduction of the necessary sieving. The paper saves a log factor over Montgomery/Weinberger, so the sievelimit is up to $2^{52}$ typically, better by 1000x from previous approaches (Arno et.al. and Wagner). The range from $2^{162}$ to $e^{260000000}$ or something (the paper has two different numbers at places) is mechanical (5 hours) from low-lying zeros of auxiliary $L$-functions; the range from $2^{52}$ to $2^{162}$ also goes this way, though not easily. He says 100000x harder to do $h\le 1000$. |
Jun
5 |
revised |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Insert FI ref to Deligne 2-variable |
Jun
5 |
answered | Good uses of Siegel zeros? |
Jun
5 |
comment |
Heuristic for Montgomery's conjecture
I think Katz has something in his book on Kloostermann sums, that they are equidistributed in some suitable sense (look at all Kloos to given a modulus, similar to looking at all characters). This is then a function field analogue perhaps. Maybe this is a bit of stretch though. |
Jun
5 |
revised |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
minor fixes, explain Weyl more, and trivial estimate |
Jun
4 |
comment |
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
"On the other hand, perhaps the connection with Elliot-Halberstam was previously under investigated considering that any result would have been conditional." $$ $$ I seriously dispute with this comment, as I think the whole field was pretty much discerned. GPY in particular made extensive effort to apply to $p_{n+\nu}-p_n$. GPY were also quite extensive in their results from partial EH, and indeed the intersection. (See Theorem 3 in Annals paper, one gets $(\sqrt\nu-\sqrt{2\theta})^2$ where $\theta$ is the distro level, and an extra factor of $e^{-\gamma}$ from Maier matrix later). |
Jun
4 |
revised |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
added 3816 characters in body |