PARI/GP actually has a better convention : only even Bernoulli numbers exist
? bernvec(5) %1 = [1, 1/6, -1/30, 1/42, -1/30, 5/66] And the two conventions can be recovered as evaluations of Bernoulli polynomials at 0 and 1 respectively ? [subst(bernpol(n), x, 0) | n <- [1..6]] %2 = [-1/2, 1/6, 0, -1/30, 0, 1/42] ? [subst(bernpol(n), x, 1) | n <- [1..6]] %3 = [1/2, 1/6, 0, -1/30, 0, 1/42] I second Edgar that I don't see much of the point on doing this change. If we had to do any change I would suggest to raise an error when bernoulli(n) is called with n=1 so that nobody could complain about sage convention choices. Vincent On Tue, 13 Sept 2022 at 04:00, edgardi...@gmail.com <edgardiasco...@gmail.com> wrote: > > The choice of the sign is arbitrary. So why make this change? What is the > benefit? > Most recent papers, algebra systems (Maple/Mathematica/Magma/Matlab/Oscar), > and libraries (Pari/Flint/Mpmath/ARB) seemed to have picked B_1 = -1/2. > Thus why put work into changing the default value and go against the current > norm? > By doing this you will generate a lot of unnecessary work to go from one > arbitrary choice (that most people use) to another that few seem to use. > On Monday, September 12, 2022 at 5:09:24 PM UTC-4 Fredrik Johansson wrote: >> >> I'm pretty neutral about this change, but I've received PRs for FLINT and >> mpmath (presumably there will be one for Arb as well) so I suppose I will >> need to make a decision about merging or closing them sooner or later :-) >> >> The sign convention for B_1 is fairly arbitrary, and the downside of >> changing it is that this introduces ambiguity and inconsistency where there >> was none before. On the other hand, you can make the case that "patching" >> deficient conventions is better in the long run... at least if the new >> convention eventually sees near-universal adoption (which is not at all >> guaranteed here). >> >> An advantage of B+ is that it would agree with the definition used in Sage >> for generalized Bernoulli numbers when restricted to the trivial character; >> see >> https://doc.sagemath.org/html/en/reference/modmisc/sage/modular/dirichlet.html#sage.modular.dirichlet.DirichletCharacter.bernoulli >> >> The claim "bernoulli_plus admits a natural generalisation to real and >> complex numbers but bernoulli_minus does not" (made elsewhere in this >> thread) seems a bit hyperbolic. For B+ this natural generalization is >> -n*zeta(1-n); for B- one can just use -n*zeta(1-n)*cos(pi*n). OK, one is a >> bit simpler than the other, but both are perfectly fine entire functions. >> >> For Sage and mpmath, I suppose the least intrusive option is to introduce a >> keyword argument to select the plus/minus convention, with B- as default >> until there is community consensus to change it. There is not really a >> strong need to change low-level libraries like FLINT at this point since >> it's trivial to handle both conventions in a wrapper. >> >> Fredrik >> >> >> On Saturday, September 10, 2022 at 4:17:08 PM UTC+2 redde...@gmail.com wrote: >>> >>> My name is Jeremy Tan, or Parcly Taxel in the furry/MLP art scene. As of >>> this post I am a recent graduate from the National University of Singapore >>> with two degrees in maths and computer science. >>> >>> Over the past month I had a good read of Peter Luschny's Bernoulli >>> Manifesto (http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html) and >>> was thoroughly convinced that B_1 (the first Bernoulli number) has to be >>> +½, not -½. (Much of Luschny's argument centres on being able to (1) >>> interpolate the Bernoulli numbers when B_1 = +½ with an entire function >>> intimately related to the zeta function, and (2) extend the range of >>> validity of or simplify several important equations like the >>> Euler–Maclaurin formula. Have a read yourself though – it is close to >>> divine truth.) >>> >>> So I went to SymPy – one of SageMath's dependencies, and where a discussion >>> on this topic was open (https://github.com/sympy/sympy/issues/23866) – and >>> successfully merged several PRs there >>> (https://github.com/sympy/sympy/pull/23926) implementing both that change >>> and some functions in Luschny's "An introduction to the Bernoulli function" >>> (https://arxiv.org/abs/2009.06743). >>> >>> I thought I was also done with changing B_1 = +½ for SageMath, but then >>> someone pointed out that the latter currently uses other libraries that all >>> have B_1 = -½. I have already opened a PR for one such library, FLINT, to >>> change B_1 = +½ there (https://github.com/wbhart/flint2/pull/1179). However >>> Fredrik Johansson has advised me that I take the discussion right here, to >>> sage-devel, because (in his words) >>> >>> > if FLINT and Arb change their definitions but the Sage developers decide >>> > that they don't like it, they will just treat the new behavior as a bug >>> > and add a special case in the wrapper to return B_1 = -½. >>> >>> So my proposal is to special-case it the other way – before the backend >>> selection in Sage's Bernoulli code >>> (https://github.com/sagemath/sage/blob/08202bc1ba7caea46327908db8e3715d1adf6f9a/src/sage/arith/misc.py#L349), >>> add a check for argument 1 and immediately return +½ if that is the case. >>> This also has the advantage of bypassing libraries that haven't or don't >>> want to change. >>> >>> What do you think? >>> >>> Jeremy Tan / Parcly Taxel > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-devel/9c61b1ee-d2c4-4c6c-b539-f89ad6d45941n%40googlegroups.com. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. 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