It is not the same, but Chowla then Rosser chased around vaguely similar ideas (weighted sums) to show that $L(s,\chi)>0$ for real $s>0$ for various real $\chi$. Chowla: matwbn.icm.edu.pl/ksiazki/aa/aa1/aa119.pdf Rosser: ams.org/journals/bull/1949-55-10/S0002-9904-1949-09306-0/…

My favorite Magma bug was when NthPrime(4) was 6. Supposedly, with NthPrime they implemented a system involving checkpoints and $x/log(x)$ estimations for $x\ge 10$, and then hard-coded the first few primes as: 2, 3, 5, 6. Oops...

I personally agree wholeheartedly here. Science grows on independent confirmation, as almost any part of the supply line can fail (software, compilers, hardware). As Mike Rubinstein put it (I think): Why do we trust software to compute zeros of $L$-functions? Well, we keep tweaking the program (that is, fixing bugs) until it gives $14.1347...$ for the Riemann $\zeta$-function, and then the same for all other cases... In other words, the answer it "correct" when it matches our expected reality. :)

First a note: due to its concept, SAGE can also inherit many of the bugs of its parts (PARI, and more). I've seen SAGE Days talks where they try to ferret out symbolic calculus contingencies, but, to be blunt, it seemed that the intersection of those capable addressing the problems with those interested in doing it (typed expressions to start) in SAGE, was zero, at least then. For an alternate count, Magma does list bug-fixes in patch releases. For version 2.15 over a year, there were 250 fixes listed. Magma doesn't do much analysis, and has a low interface, so it avoids all those worries.

Also, this is longer not shorter, but the use of Eisenstein series to prove non-vanishing on the 1-line was in vogue (originally Jacquet-Shalika in the 70s springerlink.com/index/H626367320663544.pdf ), due to Sarnak's reworking a few years ago. See math.huji.ac.il/~erezla/papers/steverevised.pdf (and the Sarnak ref of there), which does a zero-free region. The proof that symmetric square $L$-functions don't vanish at the edge, and indeed lack Siegel zeros (unless induced) by Goldfeld, Hoffstein, Lieman followed the product idea. jstor.org/stable/2118544

I can let Robin Chapman speak for himself for the illusion, but from the real-complex idea, you are using one extra $\zeta(s)$ when $\chi$ is complex (and so, overkill in pretending it is hard), and taking $(\zeta(s)L(s,\chi))^2$ when $\chi$ is real (and so, using the square of what can be used, and so for no reason I can determine, other than to uniformize the language).

I guess I can re-phrase my comment as: if you are going to use the class number formula in the end, it seems an excess to do so with cyclotomic fields. As Robin Chapman noted, you can just pair off complex conjugate characters quite easily, and then are left with just quadratics for the class number formula (which are easier from a ground-up viewpoint, though for someone facile in number fields, it matters not I guess). Of course, this is notably opposite to the what the OP wanted.

First a quibble: is it common to take the $\zeta$-function of a cyclotomic ring, rather than a cyclotomic field? And #1 should say "ideals of Z[zeta_n]"? Is it that easy to declare (to a student) that $Z(s)$ is indeed the $\zeta$-function of the cyclotomic field -- coming from an analytic number theory background (Ayoub's book from the 60s for me actually), Kronecker-Weber, or whatever Lang would have in his book, might end up being a side-trip if this is not obvious. For that matter, discreteness of the unit group (as compared to class no formula for quadratic chars) is also overhead.

I definitely agree that it is an illusion for the lack of real-complex distinction. I really like your re-phrasing as "pretends the complex case is as hard as the real case", which can be said about almost the attempts given here.

As for (1), I think it is a bad idea. There is a fundamental distinction between the two cases. The idea is, that if $L(1,\chi)=0$ for $\chi$ complex, then so does its conjugate, which gives a double zero in the product that David Speyer gives below. Enlarging this idea, we get a fairly decent zero-free region for $L(1,\chi)=0$ for $\chi$ complex (I don't recall, but $1/\log D$ maybe). But for real characters this is not true, and we have to worry about the so-called Siegel zeros, and the zero-free region is much worse (as $1/\sqrt D$). The difference is the single vs double effect.

I have heard that Allan Steel has a version of F5 in Magma, which works better than F4 in some specific examples, but it is not publically available. I definitely agree that GB algos are extremely tricky to implement well. My notion is that F4 has been tried by a few people (maybe 10 in all) to implement it, but only Faugere and Steel consistently can beat a souped-up version of the classical Buchberger. Another idea for GBs would be to take something already out there, and widen the set of base rings allowed. Inexact rings, are a real pain for correctness. but would be interesting.

Doesn't SAGE just out-source most algebraic geometry to SINGULAR or Macaulay2? I personally would say that polyhedral homotopy continuation (PHoM, PHCPack, HOM4PS, Bertini) is the place to go, though it seems that the automotive industry (robotics) has been the leader in implementing much of it.

"Cohen told me that he and Lenstra were in the audience at one of Apery's first public lectures about his proof. He said that they were madly scribbling during the talk, and by the end they were convinced." I heard Lenstra did real-time numerics on his calculator to verify some claims, and gave them greater confidence to delve further. As vdP says, they realised the crux later (despite being "convinced" previously), and was unproved (to them) prior to Zagier. I dont know if Apery stressed where the difficulties were, or if he was asked how it worked. A=B for identities wasnt known widely then

"Upon re-reading it and the article that Wadim linked it became clear that the so-called "community" acted in a worst possible manner. It was only thanks to the determination of a few outstanding mathematicians that he got the recognition that he deserved." I would say this differently. Thru the determination of others Apery's ideas went from a convoluted multi-100 page work to a 3-page note. Has anyone bothered to see if his original manuscript did in fact prove bounded denominators? That's the crux, and vDP's cheeky "utterly compelling" numerically (so is 1.2020569..., no?) is a nip off.

One "idea" of the Connes reformulation is that one can "see" how a dynamical system of primes could prove RH, if one ignores issues like renormalisation and dealing with infinitely many primes rather than finitely many (he proves the S-local analogue of the trace formula). His later programme with Marcolli/Consani has used evermore iffy language and analogues, IMO. On the mathematical end, Meyer had a nice paper on some of the function space constructs, though he doesn't actually try to get RH to appear via his re-working. projecteuclid.org/euclid.dmj/1113847338

The idea that a curve of rank 0 has all its rational points as integral is dependent on the model. See 14a2: $y^2 + xy + y = x^3 - 36x - 70$ and note $(-9/4,5/8)$ though the rank is 0.

There are other ways to show rank 0. One is to compute the analytic rank as 0 (via L-series or modular symbols) and apply Kolyvagin. A worry with SAGE, or at least PARI used by SAGE, is that it automatically assumes a GRH-like bound when computing class numbers. This is not a problem if you use Cremona's invariant based method to compute covering curves, but you need to read the details closely. I don't know which is the default in SAGE, or how to control it as an option (Simon's method). In the case where the curve has a 2-isogeny, it might use a different process altogether.

It seems that the experts say that it follows from Siegel's 1968 paper. Berechnung von Zetafunktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Goettingen Math. Phys. Kl. II, 10 (1969), 87--102. Deligne's Corvallis paper (section 6) says that his conjecture is true here due to Siegel. The main "write-up" of how Siegel applies to this is Theorem 1.2 (page 504) of Coates-Lichtenbaum? jstor.org/stable/1970916

In this list, for instance, Stein fails to mention equidistribution results. Sarnak's book (1990) "Some applications of modular forms" has these (chapter 2), and includes the "wackier" Ramanujan graphs (chapter 3) too. Also, denoting Stein's book as "new" made me think he had something more recent than Feb 2007. Sigh.