Thank you for coming, Luschny.
I not only wholeheartedly believe B_1 = +½ and that there is no convention
about it, but also that I believe in the reality and usefulness of the
Bernoulli, Euler and other functions you defined in your 2020 paper.
When I implemented those functions in SymPy there was already a base –
two-argument bernoulli() and euler() with polynomial support were there, as
well as genocchi(). So I could easily generalise those functions into yours.
On the other hand, until writing this reply I did not know that SageMath
exposed a hurwitz_zeta() function by default from which I can define the
generalised Bernoulli function. Hold on; I'll have a new ticket ready.
Jeremy Tan / Parcly Taxel
On Thu, 15 Sept 2022, 02:01 Peter Luschny, <peter.lusc...@gmail.com> wrote:
> tl;dr, 80 lines, sorry.
>
> I think there is a better alternative than changing the current
> implementation of the Bernoulli numbers.
>
> Fredrik: "The sign convention for B_1 is fairly arbitrary, ..."
>
> Calling the question a 'convention' sets a wrong framing from
> the start. Conventions are treated differently than bugs.
> And this is a bug. Setting B- leads to inconsistency in several
> identities, which I have described on my known web-page.
>
> Fredrik, honestly, do you really think Knuth rewrites TAOCP
> and CM just to change an arbitrary sign convention? Knuth
> expressly considers it a mistake.
>
> As long as the discussion is viewed as one about conventions,
> it is uninteresting and does not improve our understanding of
> the situation. People who believe this can equally well flip
> a coin when in doubt.
>
> Best leave the whole discussion behind, the question of + or -,
> the question of convention or bug, the question of decision
> or compromise, or what people on Twitter think (Fredrik asked :))
> All answers to such questions must be profoundly unsatisfactory.
>
> The matter should be solved mathematically in a way Euler
> would have liked, without resorting to this absurd fixation
> on a single value.
>
> And by that, I mean to define a suitable complex function,
> call it the 'Bernoulli function', and then say: The 'Bernoulli
> numbers' are the values of this function on the non-negative integers.
>
> Everyone knows a possible way to do this: B(z) = -z*zeta(1-z).
> Of course, you could add a sin or cos factor to it (Fredrik
> mentioned that, Knuth too, by the way), but why should you do
> that? Occam's razor speaks against it. Maximum simplicity has
> always belonged to the beauty criteria in mathematics.
>
> Similar approaches have already been done; Helmut Hasse, for
> example, has constructed the corresponding function without
> recourse to the zeta function. And there are also other
> representations without reference to the zeta function.
>
> Another pleasing possibility is via the Jensen-Johansson-Blagouchine
> formulas, first given by Jensen and first proved by Johansson
> and Blagouchine ("Computing Stieltjes constants using complex
> integration").
>
> These formulas are also the starting point in my arXiv paper.
> There I try to show that this function is also essential in
> other contexts in the theory of special numbers and offers
> some technical advantages.
>
> * My suggestion: Offer this function in Sage. This may not mean
> much more than a three-line wrapper around the zeta function.
> Then the current code does not have to be changed, no deprecation
> warnings, no keywords for alternatives are required.
>
> The decision is then solely up to the user whether he wants
> to continue the current usage and use bernoulli(n) or find
> pleasure in bernoulli_function(n).
>
> Fredrik's fear that new "ambiguity and inconsistency" could
> creep in is then unfounded: The names and the definitions are
> too different; here polynomials, there a zeta-like function,
> both with different applications.
>
> If you want to describe this approach in a highfalutin way,
> it is Grothendieck-ish. It does not answer the question which
> of the two values is the 'correct' one; it shows how this problem
> disappears when put into the proper conceptual framework.
>
> This approach also makes sense in all the other cases that
> Fredrik has to decide. With it, the burden of making a sensible
> decision is turned into the freedom for the user to explore
> a fascinating function.
>
> --
> Peter
>
>
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