Bernoulli(1) = +½

[Jeremy Tan]

I am trying to get various mathematical software libraries to change the return value at 1 of their implementation of the Bernoulli numbers to +½. I've already done it for SymPy, and now want FLINT and Arb to change too. The relevant PR for FLINT, with a brief discussion on the merits of this change, is here:

https://github.com/wbhart/flint2/pull/1179

I'd like to have this merged as soon as possible and to have a similar change made for Arb.

Parcly Taxel

[Fredrik Johansson]

I'm not opposed in principle to the new definition. For example, a nice feature of having B_1 = +1/2 is that it agrees with the most common definition of generalized Bernoulli numbers when specialized to the trivial character.

However, this would be inconsistent with Pari/GP, Magma, Maple, Mathematica, etc., not to mention several decades of mathematical literature. It is unlikely that all those other systems will change their definition, and having two conventions around is arguably worse than having a single convention that is slightly less elegant. It's maybe not really a big deal since most applications only use the even-index Bernoulli numbers, but I'm sure this will break *some* user code.

Probably this should be brought up on sage-devel? It's not enough to change the libraries if you want to get this into Sage; 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 = -1/2.

Fredrik

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[Jeremy Tan]

As a compromise between Luschny's (and my) views and those who want B_1 to stay as -½, I've sent in a PR to Arb implementing Luschny's (generalised) Bernoulli function, but not touching the Bernoulli number code:

https://github.com/fredrik-johansson/arb/pull/438

I might need your help with the tests and other little formalities.

Jeremy Tan / Parcly Taxel