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/1b4bcfca-5200-4514-b082-e03dbab004bcn%40googlegroups.com.
